solving statistics probability problems

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Teach yourself statistics

Statistics Problems

One of the best ways to learn statistics is to solve practice problems. These problems test your understanding of statistics terminology and your ability to solve common statistics problems. Each problem includes a step-by-step explanation of the solution.

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Main topic:

Problem description:

In one state, 52% of the voters are Republicans, and 48% are Democrats. In a second state, 47% of the voters are Republicans, and 53% are Democrats. Suppose a simple random sample of 100 voters are surveyed from each state.

What is the probability that the survey will show a greater percentage of Republican voters in the second state than in the first state?

The correct answer is C. For this analysis, let P 1 = the proportion of Republican voters in the first state, P 2 = the proportion of Republican voters in the second state, p 1 = the proportion of Republican voters in the sample from the first state, and p 2 = the proportion of Republican voters in the sample from the second state. The number of voters sampled from the first state (n 1 ) = 100, and the number of voters sampled from the second state (n 2 ) = 100.

The solution involves four steps.

• Make sure the sample size is big enough to model differences with a normal population. Because n 1 P 1 = 100 * 0.52 = 52, n 1 (1 - P 1 ) = 100 * 0.48 = 48, n 2 P 2 = 100 * 0.47 = 47, and n 2 (1 - P 2 ) = 100 * 0.53 = 53 are each greater than 10, the sample size is large enough.
• Find the mean of the difference in sample proportions: E(p 1 - p 2 ) = P 1 - P 2 = 0.52 - 0.47 = 0.05.

σ d = sqrt{ [ P1( 1 - P 1 ) / n 1 ] + [ P 2 (1 - P 2 ) / n 2 ] }

σ d = sqrt{ [ (0.52)(0.48) / 100 ] + [ (0.47)(0.53) / 100 ] }

σ d = sqrt (0.002496 + 0.002491) = sqrt(0.004987) = 0.0706

z p 1 - p 2 = (x - μ p 1 - p 2 ) / σ d = (0 - 0.05)/0.0706 = -0.7082

Using Stat Trek's Normal Distribution Calculator , we find that the probability of a z-score being -0.7082 or less is 0.24.

Therefore, the probability that the survey will show a greater percentage of Republican voters in the second state than in the first state is 0.24.

See also: Difference Between Proportions

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How to Calculate Probability

Last Updated: July 31, 2023 Fact Checked

This article was co-authored by Mario Banuelos, PhD . Mario Banuelos is an Assistant Professor of Mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical models for genome evolution, and data science. Mario holds a BA in Mathematics from California State University, Fresno, and a Ph.D. in Applied Mathematics from the University of California, Merced. Mario has taught at both the high school and collegiate levels. There are 8 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 3,371,109 times.

Chances are (pun intended) you've encountered probability by now, but what exactly is probability, and how do you calculate it? Probability is the likelihood of a specific event happening, like winning the lottery or rolling a 6 on a die. Finding probability is easy using the probability formula (the number of favorable outcomes divided by the total number of outcomes). In this article, we'll walk you through exactly how to use the probability formula step by step, plus show you some examples of the probability formula in action.

Finding the Probability of a Single Random Event

Example: It would be impossible to calculate the probability of an event phrased as: “Both a 5 and a 6 will come up on a single roll of a die.”

• Example 1 : What is the likelihood of choosing a day that falls on the weekend when randomly picking a day of the week? "Choosing a day that falls on the weekend" is our event, and the number of outcomes is the total number of days in a week: 7.
• Example 2 : A jar contains 4 blue marbles, 5 red marbles and 11 white marbles. If a marble is drawn from the jar at random, what is the probability that this marble is red? "Choosing a red marble" is our event, and the number of outcomes is the total number of marbles in the jar, 20.

• Example 1 : What is the likelihood of choosing a day that falls on the weekend when randomly picking a day of the week? The number of events is 2 (since 2 days out of the week are weekends), and the number of outcomes is 7. The probability is 2 ÷ 7 = 2/7. You could also express this as 0.285 or 28.5%.
• Example 2 : A jar contains 4 blue marbles, 5 red marbles and 11 white marbles. If a marble is drawn from the jar at random, what is the probability that this marble is red? The number of events is 5 (since there are 5 red marbles), and the number of outcomes is 20. The probability is 5 ÷ 20 = 1/4. You could also express this as 0.25 or 25%.

• For example, the likelihood of rolling a 3 on a 6-sided die is 1/6. But the probability of rolling all five other numbers on a die is also 1/6. 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 6/6 , which = 100%.

Note: If you had, for example, forgotten about the number 4 on the dice, adding up the probabilities would only reach 5/6 or 83%, indicating a problem.

• For example, if you were to calculate the probability of the Easter holiday falling on a Monday in the year 2020, the probability would be 0 because Easter is always on a Sunday.

Calculating the Probability of Multiple Random Events

Note: The probability of the 5s being rolled are called independent events, because what you roll the first time does not affect what happens the second time.

• Now, the likelihood that the second card is a club is 12/51, since 1 club will have already been removed. This is because what you do the first time affects the second. If you draw a 3 of clubs and don't put it back, there will be one less club and one less card in the deck (51 instead of 52).
• The probability that the first marble is red is 5/20, or 1/4. The probability of the second marble being blue is 4/19, since we have 1 less marble, but not 1 less blue marble. And the probability that the third marble is white is 11/18, because we’ve already chosen 2 marbles.

• Example 1 : Two cards are drawn randomly from a deck of cards. What is the likelihood that both cards are clubs? The probability of the first event happening is 13/52. The probability of the second event happening is 12/51. The probability is 13/52 x 12/51 = 12/204 = 1/17. You could also express this as 0.058 or 5.8%.
• Example 2 : A jar contains 4 blue marbles, 5 red marbles and 11 white marbles. If three marbles are drawn from the jar at random, what is the probability that the first marble is red, the second marble is blue, and the third is white? The probability of the first event is 5/20. The probability of the second event is 4/19. And the probability of the third event is 11/18. The probability is 5/20 x 4/19 x 11/18 = 44/1368 = 0.032. You could also express this as 3.2%.

Converting Odds to Probabilities

• The number 11 represents the likelihood of choosing a white marble and the number 9 represents the likelihood of choosing a marble of a different color.
• So, odds are that you will draw a white marble.

• The event that you’ll draw a white marble is 11; the event another color will be drawn is 9. The total number of outcomes is 11 + 9, or 20.

• So, in our example, the probability of drawing a white marble is 11/20. Divide this out: 11 ÷ 20 = 0.55 or 55%.

Probability Cheat Sheets

Expert Q&A

Video . By using this service, some information may be shared with YouTube.

• Mathematicians typically use the term “relative probability” to refer to the chances of an event happening. They insert the word "relative" since no outcome is 100% guaranteed. For example, if you flip a coin 100 times, you probably won't get exactly 50 heads and 50 tails. Relative probability takes this caveat into account. [10] X Research source Thanks Helpful 0 Not Helpful 1
• You may need to know that that in sports betting and bookmaking, odds are expressed as “odds against,” which means that the odds of an event happening are written first, and the odds of an event not happening come second. Although it can be confusing, it's important to know this if you’re planning to bet on a sporting event. Thanks Helpful 12 Not Helpful 4
• The most common ways of writing down probabilities include putting them as fractions, as decimals, as percentages, or on a 1–10 scale. Thanks Helpful 8 Not Helpful 5

You Might Also Like

• ↑ https://www.theproblemsite.com/reference/mathematics/probability/mutually-exclusive-outcomes
• ↑ Mario Banuelos, PhD. Assistant Professor of Mathematics. Expert Interview. 11 December 2021.
• ↑ https://www.mathplanet.com/education/pre-algebra/probability-and-statistic/probability-of-events
• ↑ https://www.mathsisfun.com/probability_line.html
• ↑ https://www.probabilisticworld.com/not-all-zero-probabilities/
• ↑ https://www.khanacademy.org/math/ap-statistics/probability-ap/stats-conditional-probability/e/calculating-conditional-probability
• ↑ https://www.mathsisfun.com/data/probability.html
• ↑ https://www.mathsisfun.com/data/probability-events-types.html

About This Article

Probability is the likelihood that a specific event will occur. To calculate probability, first define the number of possible outcomes that can occur. For example, if someone asks, “What is the probability of choosing a day that falls on the weekend when randomly picking a day of the week,” the number of possible outcomes when choosing a random day of the week is 7, since there are 7 days of the week. Now define the number of events. In this example, the number of events is 2 since 2 days out of the week fall on the weekend. Finally, divide the number of events by the number of outcomes to get the probability. In our example, we would divide 2, the number of events, by 7, the number of outcomes, and get 2/7, or 0.28. You could also express the answer as a percentage, or 28.5%. Therefore, there’s a 28.5% probability that you would choose a day that falls on the weekend when randomly picking a day of the week. To learn how to calculate the probability of multiple events happening in a row, keep reading! Did this summary help you? Yes No

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Probability Questions with Solutions

Tutorial on finding the probability of an event. In what follows, S is the sample space of the experiment in question and E is the event of interest. n(S) is the number of elements in the sample space S and n(E) is the number of elements in the event E.

Questions and their Solutions

Answers to the above exercises, more references and links.

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How to Solve Probability Problems? (+FREE Worksheet!)

Do you want to know how to solve Probability Problems? Here you learn how to solve probability word problems.

Related Topics

• How to Interpret Histogram
• How to Interpret Pie Graphs
• How to Solve Permutations and Combinations
• How to Find Mean, Median, Mode, and Range of the Given Data

Step by step guide to solve Probability Problems

• Probability is the likelihood of something happening in the future. It is expressed as a number between zero (can never happen) to \(1\) (will always happen).
• Probability can be expressed as a fraction, a decimal, or a percent.
• To solve a probability problem identify the event, find the number of outcomes of the event, then use probability law: \(\frac{number\ of \ favorable \ outcome}{total \ number \ of \ possible \ outcomes}\)

Probability Problems – Example 1:

If there are \(8\) red balls and \(12\) blue balls in a basket, what is the probability that John will pick out a red ball from the basket?

There are \(8\) red balls and \(20\) a total number of balls. Therefore, the probability that John will pick out a red ball from the basket is \(8\) out of \(20\) or \(\frac{8}{8+12}=\frac{8}{20}=\frac{2}{5}\).

Probability Problems – Example 2:

A bag contains \(18\) balls: two green, five black, eight blue, a brown, a red, and one white. If \(17\) balls are removed from the bag at random, what is the probability that a brown ball has been removed?

If \(17\) balls are removed from the bag at random, there will be one ball in the bag. The probability of choosing a brown ball is \(1\) out of \(18\). Therefore, the probability of not choosing a brown ball is \(17\) out of \(18\) and the probability of having not a brown ball after removing \(17\) balls is the same.

Exercises for Solving Probability Problems

• A number is chosen at random from \(1\) to \(10\). Find the probability of selecting a \(4\) or smaller.
• A number is chosen at random from \(1\) to \(50\). Find the probability of selecting multiples of \(10\).
• A number is chosen at random from \(1\) to \(10\). Find the probability of selecting of \(4\) and factors of \(6\).
• A number is chosen at random from \(1\) to \(10\). Find the probability of selecting a multiple of \(3\).
• A number is chosen at random from \(1\) to \(50\). Find the probability of selecting prime numbers.
• A number is chosen at random from \(1\) to \(25\). Find the probability of not selecting a composite number.

Download Probability Problems Worksheet

• \(\color{blue}{\frac{2}{5}}\)
• \(\color{blue}{\frac{1}{10}}\)
• \(\color{blue}{\frac{1}{2}}\)
• \(\color{blue}{\frac{3}{10}}\)
• \(\color{blue}{\frac{9}{25}}\)

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How to Solve Probability Problems in Statistics

Individuals are looking for online or offline classes to help them understand the basics of probability in statistics. The reason for this might be that they confuse the instructions applied to probability as there are many . One might be confused with the additions, multiplications, and combinations. One can identify when they were studying probability, and also after they passed the course. One still struggles with the tails and heads of estimating while to practice which rule. This post has given a review of everyday conditions that will explain the methods for how to solve probability problems in statistics utilizing the right system.

The models of probability queries presented are uncomplicated cases, such as the benefits of selecting something or gaining the things. Later on, one can come over probability distributions such as the normal distribution and the binomial distribution. One can normally acknowledge they are working on a probability distribution query by keywords such as “fits a binomial distribution” or “normally distributed.” If this is the case, one can verify the probability index for the various posts about probability problems that include different distributions.

Methods required to solve probability problems

Table of Contents

Get the keyword. This is one of the important tips to solve the probability term problem that involves getting the keyword. This will help the learners to recognize which theorem is used for solving the probability problems. The keywords can be “or” “and” and “not.” For example, suppose this word query: “Find out the probability that Sam will select both the vanilla and chocolate ice cream delivered that he will select vanilla 60% of the time, chocolate 70% of the time, and none of the 10% of the time.” The query holds the keyword “and.”

Decide which the functions are commonly independent or exclusive, if suitable. While utilizing a rule of multiply, one has two choices to select from. One can use the theorem P(A and B) = P(A) x P(B) while the possibilities A and B are unconventional. One applies the rule P(A and B) = P(A) x P(B|A) while the chances are subjective. P(B|A) is a conditional probability, which means that event A happens when event B has previously happened. 

Get the separate components of the given equation. Any probability equation has several elements that require it to be chosen to resolve the query. For instance, one can learn the keyword “and” and then apply the rule to practice a multiplication rule. As the events do not depend on the other event, one can apply the rule P(A and B) = P(A) x P(B). The action initiates P(A) = probability of event A happening and P(B) = probability of event B. The query states that P(A = vanilla ) = 60% and P(B = chocolate ) = 70%.

Change the contents in the supplied equation. One can change the word “vanilla” while seeing the event A and the word “chocolate” while one will see the event B. Applying the relevant equation for the model and changing the values will now be P(vanilla and chocolate) = 60% x 70%.

Answer the given equation. Practice the earlier model, P(vanilla and chocolate) = 60%x 70%. Separating the percentages value in decimals will produce 0.60 x 0.70 that is determined by sorting both percentages value by 100—the multiplication events into the value will be 0.42. Changing the result into a percentage with multiplying the value with 100 that will produce 42%.

Now take some of the other examples of how to solve probability problems in statistics.

How to solve probability problems in statistics about events

Determining the sample events’ probability that is occurring is honestly: sum the possibilities together. For instance, if one has a 20% possibility of obtaining $20 and a 35% possibility of winning $30, the overall probability of obtaining something will be 20% + 35% = 55%. It only operates for commonly independent events (events that are not occurring at the equivalent time).

How to solve probability problems in statistics for dice rolling

One can use one dice to resolve the dice rolling questions, or they can use three dice. The probability modifies based on how much quantity of dice one is rolling and what numeric value they want to select. The quickest method to solve these kinds of probability questions is to figure out all the feasible dice sequences (this is known as writing the sample space). Here, we have mentioned very easy example, if one likes to check the probability of double dice’s rolling, the sample space could be:

[1][1], [1][2], [1][3], [1][4], [1][5], [1][6],

[2][1], [2][2], [2][3], [2][4],[2][5], [2][6],

[3][1], [3][2], [3][3], [3][4], [3][5], [3][6],

[4][1], [4][2], [4][3], [4][4], [4][5], [4][6],

[5][1], [5][2], [5][3], [5][4], [5][5], [5][6],

[6][1], [6][2], [6][3], [6][4], [6][5], [6][6].

You want to check doubles; then you can see that there are six sequences: [1][1], [2][2], [3][3], [4][4], [5][5], [6][6] out of 36 possible lists; therefore, the probability will be 6/36. Students can utilize the corresponding sample term to decide all odds of dice rolling a 2 and a 3 (2/36), or this is the two dice sum as 7. In the above case, the 7 will be the sum of: [6][1], [1][6], [3][4], [4][3], [5,2], [2,5] that is why the probability will be 6/36. This is how to solve probability problems in statistics.

How to solve probability problems in statistics using cards

One can practice a similar method utilized for rolling the dice (view above): Record all the possible sample space. For an individual regular deck of cards, one has 52 cards. The sample space will be:

• clubs: J, Q, K, A, 2, 3, 4, 5, 6, 7, 8, 9, 10
• hearts: J, Q, K, A, 2, 3, 4, 5, 6, 7, 8, 9, 10
• diamonds: J, Q, K, A, 2, 3, 4, 5, 6, 7, 8, 9, 10
• spades: J, Q, K, A, 2, 3, 4, 5, 6, 7, 8, 9, 10

If a person were to claim the probability of selecting a club or a 2, it would be 13 clubs (with the 2 of clubs) and the other three “2”s, giving 16 cards. Therefore, the probability will be 16/52 or 4/13.

These are the three different examples that are used to recognize the methods for how to solve probability problems in statistics.

Conclusion 

To sum up the post on how to solve probability problems in statistics, we can say that three different methods can be used to solve them. Besides these methods, there are numerous problems that can be solved by learners. Therefore, try to remember these methods and avoid them while solving probability. Probability has significant uses in day to day lives that are beneficial to solve various daily problems. So, learn the methods to solve probability problems and get the benefits of these to overcome daily numeric problems. Moreover, if you have any problem related to statistics assignment, then you can hire our Statistics assignment helper and get the best help with statistics assignment from the experts.

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Statistics and probability

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Statistics and Probability Worksheets

Welcome to the statistics and probability page at Math-Drills.com where there is a 100% chance of learning something! This page includes Statistics worksheets including collecting and organizing data, measures of central tendency (mean, median, mode and range) and probability.

Students spend their lives collecting, organizing, and analyzing data, so why not teach them a few skills to help them on their way. Data management is probably best done on authentic tasks that will engage students in their own learning. They can collect their own data on topics that interest them. For example, have you ever wondered if everyone shares the same taste in music as you? Perhaps a survey, a couple of graphs and a few analysis sentences will give you an idea.

Statistics has applications in many different fields of study. Budding scientists, stock market brokers, marketing geniuses, and many other pursuits will involve managing data on a daily basis. Teaching students critical thinking skills related to analyzing data they are presented will enable them to make crucial and informed decisions throughout their lives.

Probability is a topic in math that crosses over to several other skills such as decimals, percents, multiplication, division, fractions, etc. Probability worksheets will help students to practice all of these skills with a chance of success!

Most Popular Statistics and Probability Worksheets this Week

Mean, Median, Mode and Range Worksheets

Worksheets for students to practice finding the mean, median, mode and range or number sets.

Calculating the mean, median, mode and range are staples of the upper elementary math curriculum. Here you will find worksheets for practicing the calculation of mean, median, mode and range. In case you're not familiar with these concepts, here is how to calculate each one. To calculate the mean, add all of the numbers in the set together and divide that sum by the number of numbers in the set. To calculate the median, first arrange the numbers in order, then locate the middle number. In sets where there are an even number of numbers, calculate the mean of the two middle numbers. To calculate the mode, look for numbers that repeat. If there is only one of each number, the set has no mode. If there are doubles of two different numbers and there are more numbers in the set, the set has two modes. If there are triples of three different numbers and there are more numbers in the set, the set has three modes, and so on. The range is calculated by subtracting the least number from the greatest number.

Note that all of the measures of central tendency are included on each page, but you don't need to assign them all if you aren't working on them all. If you're only working on mean, only assign students to calculate the mean.

Determining Mean, Median, Mode and Range from Sorted lists of numbers

In order to determine the median, it is necessary to have your numbers sorted. It is also helpful in determining the mode and range. To expedite the process, these worksheets include the lists of numbers already sorted.

Determining Mean, Median, Mode and Range from Unsorted lists of numbers

Normally, data does not come in a sorted list, so these worksheets are a little more realistic. To find some of the statistics, it will be easier for students to put the numbers in order first.

Collecting and Organizing Data

Collecting and organizing data worksheets including line plots and stem-and-leaf diagrams.

Teaching students how to collect and organize data enables them to develop skills that will enable them to study topics in statistics with more confidence and deeper understanding.

Constructing line plots from smaller data sets

Constructing line plots from larger data sets

Interpreting and Analyzing Data Worksheets

Interpreting and analyzing data worksheets including worksheets with stem-and-leaf plots, line plots and various graph types.

Answering questions about graphs and other data helps students build critical thinking skills. The versions with no questions are intended for those who want to write their own questions and answers.

Questions about Stem-and-leaf plots

Standard questions include determining the minimum, maximum, range, count, median, mode, and mean.

Questions about Line plots

Questions about Broken-Line Graphs

Questions about Circle Graphs

Questions about Pictographs

Probability Worksheets

Probability worksheets including probabilities of dice and spinners with various numbers of sections.

Probability with Dice

Probability with Number Spinners

Spinners can be used for probability experiments or for theoretical probability. Students should intuitively know that a number that is more common on a spinner will come up more often. Spinning 100 or more times and tallying the results should get them close to the theoretical probability. The more sections there are, the more spins will be needed.

Probability with Non-Numerical Spinners

Non-numerical spinners can be used for experimental or theoretical probability. There are basic questions on every version with a couple extra questions on the A and B versions. Teachers and students can make up other questions to ask and conduct experiments or calculate the theoretical probability. Print copies for everyone or display on an interactive white board.

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15 Probability Questions And Practice Problems for Middle and High School: Harder Exam Style Questions Included

Beki christian.

Probability questions and probability problems require students to work out how likely it is that something is to happen. Probabilities can be described using words or numbers. Probabilities range from 0 to 1 and can be written as fractions, decimals or percentages .

Here you’ll find a selection of probability questions of varying difficulty showing the variety you are likely to encounter in middle school and high school, including several harder exam style questions.

What are some real life examples of probability?

How to calculate probabilities, probability question: a worked example, 6th grade probability questions, 7th grade probability questions, 8th grade probability questions, 9th grade probability questions, 10th grade probability questions.

• 11th & 12th grade probability questions

Looking for more middle school and high school math questions?

The more likely something is to happen, the higher its probability. We think about probabilities all the time.

For example, you may have seen that there is a 20% chance of rain on a certain day or thought about how likely you are to roll a 6 when playing a game, or to win in a raffle when you buy a ticket.

Probability Check for Understanding Quiz

Wondering if your students have fully grasped probability? Use this quiz to check their understanding across 15 questions with answers covering all things probability!

The probability of something happening is given by:

We can also use the following formula to help us calculate probabilities and solve problems:

• Probability of something not occuring = 1 – probability of if occurring P(not\;A) = 1 - P(A)
• For mutually exclusive events: Probability of event A OR event B occurring = Probability of event A + Probability of event B P(A\;or\;B) = P(A)+P(B)
• For independent events: Probability of event A AND event B occurring = Probability of event A times probability of event B P(A\;and\;B) = P(A) × P(B)

Question: What is the probability of getting heads three times in a row when flipping a coin?

When flipping a coin, there are two possible outcomes – heads or tails. Each of these options has the same probability of occurring during each flip. The probability of either heads or tails on a single coin flip is ½.

Since there are only two possible outcomes and they have the same probability of occurring, this is called a binomial distribution.

Let’s look at the possible outcomes if we flipped a coin three times.

Let H=heads and T=tails.

The possible outcomes are: HHH, THH, THT, HTT, HHT, HTH, TTH, TTT

Each of these outcomes has a probability of ⅛.

Therefore, the probability of flipping a coin three times in a row and having it land on heads all three times is ⅛.

Middle school probability questions

In middle school, probability questions introduce the idea of the probability scale and the fact that probabilities sum to one. We look at theoretical and experimental probability as well as learning about sample space diagrams and venn diagrams.

1. Which number could be added to this spinner to make it more likely that the spinner will land on an odd number than a prime number?

Currently there are two odd numbers and two prime numbers so the chances of landing on an odd number or a prime number are the same. By adding 3, 5 or 11 you would be adding one prime number and one odd number so the chances would remain equal.

By adding 9 you would be adding an odd number but not a prime number. There would be three odd numbers and two prime numbers so the spinner would be more likely to land on an odd number than a prime number.

2. Ifan rolls a fair dice, with sides labeled A, B, C, D, E and F. What is the probability that the dice lands on a vowel?

A and E are vowels so there are 2 outcomes that are vowels out of 6 outcomes altogether.

Therefore the probability is   \frac{2}{6} which can be simplified to \frac{1}{3} .

3. Max tested a coin to see whether it was fair. The table shows the results of his coin toss experiment:

Heads          Tails

    26                  41

What is the relative frequency of the coin landing on heads?

Max tossed the coin 67 times and it landed on heads 26 times.

\text{Relative frequency (experimental probability) } = \frac{\text{number of successful trials}}{\text{total number of trials}} = \frac{26}{67}

4. Grace rolled two dice. She then did something with the two numbers shown. Here is a sample space diagram showing all the possible outcomes:

What did Grace do with the two numbers shown on the dice?

Add them together

Subtract the number on dice 2 from the number on dice 1

Multiply them

Subtract the smaller number from the bigger number

For each pair of numbers, Grace subtracted the smaller number from the bigger number.

For example, if she rolled a 2 and a 5, she did 5 − 2 = 3.

5. Alice has some red balls and some black balls in a bag. Altogether she has 25 balls. Alice picks one ball from the bag. The probability that Alice picks a red ball is x and the probability that Alice picks a black ball is 4x. Work out how many black balls are in the bag.

Since the probability of mutually exclusive events add to 1:  

\begin{aligned} x+4x&=1\\\\ 5x&=1\\\\ x&=\frac{1}{5} \end{aligned}

\frac{1}{5} of the balls are red and \frac{4}{5} of the balls are blue.

6. Arthur asked the students in his class whether they like math and whether they like science. He recorded his results in the venn diagram below.

How many students don’t like science?

We need to look at the numbers that are not in the ‘Like science’ circle. In this case it is 9 + 7 = 16.

High school probability questions

In high school, probability questions involve more problem solving to make predictions about the probability of an event. We also learn about probability tree diagrams, which can be used to represent multiple events, and conditional probability.

7. A restaurant offers the following options:

Starter – soup or salad

Main – chicken, fish or vegetarian

Dessert – ice cream or cake

How many possible different combinations of starter, main and dessert are there?

The number of different combinations is 2 × 3 × 2 = 12.

8. There are 18 girls and 12 boys in a class. \frac{2}{9} of the girls and \frac{1}{4} of the boys walk to school. One of the students who walks to school is chosen at random. Find the probability that the student is a boy. 

First we need to work out how many students walk to school:

\frac{2}{9} \text{ of } 18 = 4

\frac{1}{4} \text{ of } 12 = 3

7 students walk to school. 4 are girls and 3 are boys. So the probability the student is a boy is \frac{3}{7} .

9. Rachel flips a biased coin. The probability that she gets two heads is 0.16. What is the probability that she gets two tails?

We have been given the probability of getting two heads. We need to calculate the probability of getting a head on each flip.

Let’s call the probability of getting a head p.

The probability p, of getting a head AND getting another head is 0.16.

Therefore to find p:

The probability of getting a head is 0.4 so the probability of getting a tail is 0.6.

The probability of getting two tails is 0.6 × 0.6 = 0.36 .

10. I have a big tub of jelly beans. The probability of picking each different color of jelly bean is shown below:

If I were to pick 60 jelly beans from the tub, how many orange jelly beans would I expect to pick?

First we need to calculate the probability of picking an orange. Probabilities sum to 1 so 1 − (0.2 + 0.15 + 0.1 + 0.3) = 0.25.

The probability of picking an orange is 0.25.

The number of times I would expect to pick an orange jelly bean is 0.25 × 60 = 15 .

11. Dexter runs a game at a fair. To play the game, you must roll a dice and pick a card from a deck of cards.

To win the game you must roll an odd number and pick a picture card. The game can be represented by the tree diagram below.

Dexter charges players $1 to play and gives $3 to any winners. If 260 people play the game, how much profit would Dexter expect to make?

Completing the tree diagram:

Probability of winning is \frac{1}{2} \times \frac{4}{13} = \frac{4}{26}

If 260 play the game, Dexter would receive $260.

The expected number of winners would be \frac{4}{26} \times 260 = 40

Dexter would need to give away 40 × $3 = $120 .

Therefore Dexter’s profit would be $260 − $120 = $140.

12. A fair coin is tossed three times. Work out the probability of getting two heads and one tail.

There are three ways of getting two heads and one tail: HHT, HTH or THH.

The probability of each is \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} 

Therefore the total probability is \frac{1}{8} +\frac{1}{8} + \frac{1}{8} = \frac{3}{8}

11th/12th grade probability questions

13. 200 people were asked about which athletic event they thought was the most exciting to watch. The results are shown in the table below.

A person is chosen at random. Given that that person chose 100m, what is the probability that the person was female?

Since we know that the person chose 100m, we need to include the people in that column only.

In total 88 people chose 100m so the probability the person was female is \frac{32}{88}   .

14.   Sam asked 50 people whether they like vegetable pizza or pepperoni pizza.

37 people like vegetable pizza. 

25 people like both. 

3 people like neither.

Sam picked one of the 50 people at random. Given that the person he chose likes pepperoni pizza, find the probability that they don’t like vegetable pizza.

We need to draw a venn diagram to work this out.

We start by putting the 25 who like both in the middle section. The 37 people who like vegetable pizza includes the 25 who like both, so 12 more people must like vegetable pizza. 3 don’t like either. We have 50 – 12 – 25 – 3 = 10 people left so this is the number that must like only pepperoni.

There are 35 people altogether who like pepperoni pizza. Of these, 10 do not like vegetable pizza. The probability is   \frac{10}{35} .

15. There are 12 marbles in a bag. There are n red marbles and the rest are blue marbles. Nico takes 2 marbles from the bag. Write an expression involving n for the probability that Nico takes one red marble and one blue marble.

We need to think about this using a tree diagram. If there are 12 marbles altogether and n are red then 12-n are blue.

To get one red and one blue, Nico could choose red then blue or blue then red so the probability is:

• 15 Ratio questions
• 15 Algebra questions
• 15 Trigonometry questions
• 15 Simultaneous equations questions
• 15 Venn diagram questions
• Long division questions

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The content in this article was originally written by secondary teacher Beki Christian and has since been revised and adapted for US schools by elementary math teacher Katie Keeton.

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Problem Solver Subjects

Our math problem solver that lets you input a wide variety of probability math problems and it will provide a step by step answer. This math solver excels at math word problems as well as a wide range of math subjects.

• Math Word Problems
• Pre-Algebra
• Geometry Graphing
• Trigonometry
• Precalculus
• Finite Math
• Linear Algebra

Here are example math problems within each subject that can be input into the calculator and solved. This list is constanstly growing as functionality is added to the calculator.

Basic Math Solutions

Below are examples of basic math problems that can be solved.

• Long Arithmetic
• Rational Numbers
• Operations with Fractions
• Ratios, Proportions, Percents
• Measurement, Area, and Volume
• Factors, Fractions, and Exponents
• Unit Conversions
• Data Measurement and Statistics
• Points and Line Segments

Math Word Problem Solutions

Math word problems require interpreting what is being asked and simplifying that into a basic math equation. Once you have the equation you can then enter that into the problem solver as a basic math or algebra question to be correctly solved. Below are math word problem examples and their simplified forms.

Word Problem: Rachel has 17 apples. She gives some to Sarah. Sarah now has 8 apples. How many apples did Rachel give her?

Simplified Equation: 17 - x = 8

Word Problem: Rhonda has 12 marbles more than Douglas. Douglas has 6 marbles more than Bertha. Rhonda has twice as many marbles as Bertha has. How many marbles does Douglas have?

Variables: Rhonda's marbles is represented by (r), Douglas' marbles is represented by (d) and Bertha's marbles is represented by (b)

Simplified Equation: {r = d + 12, d = b + 6, r = 2 × b}

Word Problem: if there are 40 cookies all together and Angela takes 10 and Brett takes 5 how many are left?

Simplified: 40 - 10 - 5

Pre-Algebra Solutions

Below are examples of Pre-Algebra math problems that can be solved.

• Variables, Expressions, and Integers
• Simplifying and Evaluating Expressions
• Solving Equations
• Multi-Step Equations and Inequalities
• Ratios, Proportions, and Percents
• Linear Equations and Inequalities

Algebra Solutions

Below are examples of Algebra math problems that can be solved.

• Algebra Concepts and Expressions
• Points, Lines, and Line Segments
• Simplifying Polynomials
• Factoring Polynomials
• Linear Equations
• Absolute Value Expressions and Equations
• Radical Expressions and Equations
• Systems of Equations
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• Inequalities
• Complex Numbers and Vector Analysis
• Logarithmic Expressions and Equations
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• Vector Spaces
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• Eigenvalues and Eigenvectors
• Linear Transformations
• Number Sets
• Analytic Geometry

Trigonometry Solutions

Below are examples of Trigonometry math problems that can be solved.

• Algebra Concepts and Expressions Review
• Right Triangle Trigonometry
• Radian Measure and Circular Functions
• Graphing Trigonometric Functions
• Simplifying Trigonometric Expressions
• Verifying Trigonometric Identities
• Solving Trigonometric Equations
• Complex Numbers
• Analytic Geometry in Polar Coordinates
• Exponential and Logarithmic Functions
• Vector Arithmetic

Precalculus Solutions

Below are examples of Precalculus math problems that can be solved.

• Operations on Functions
• Rational Expressions and Equations
• Polynomial and Rational Functions
• Analytic Trigonometry
• Sequences and Series
• Analytic Geometry in Rectangular Coordinates
• Limits and an Introduction to Calculus

Calculus Solutions

Below are examples of Calculus math problems that can be solved.

• Evaluating Limits
• Derivatives
• Applications of Differentiation
• Applications of Integration
• Techniques of Integration
• Parametric Equations and Polar Coordinates
• Differential Equations

Statistics Solutions

Below are examples of Statistics problems that can be solved.

• Algebra Review
• Average Descriptive Statistics
• Dispersion Statistics
• Probability
• Probability Distributions
• Frequency Distribution
• Normal Distributions
• t-Distributions
• Hypothesis Testing
• Estimation and Sample Size
• Correlation and Regression

Finite Math Solutions

Below are examples of Finite Math problems that can be solved.

• Polynomials and Expressions
• Equations and Inequalities
• Linear Functions and Points
• Systems of Linear Equations
• Mathematics of Finance
• Statistical Distributions

Linear Algebra Solutions

Below are examples of Linear Algebra math problems that can be solved.

• Introduction to Matrices
• Linear Independence and Combinations

Chemistry Solutions

Below are examples of Chemistry problems that can be solved.

• Unit Conversion
• Atomic Structure
• Molecules and Compounds
• Chemical Equations and Reactions
• Behavior of Gases
• Solutions and Concentrations

Physics Solutions

Below are examples of Physics math problems that can be solved.

• Static Equilibrium
• Dynamic Equilibrium
• Kinematics Equations
• Electricity
• Thermodymanics

Geometry Graphing Solutions

Below are examples of Geometry and graphing math problems that can be solved.

• Step By Step Graphing
• Linear Equations and Functions
• Polar Equations

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Confidence Interval for Proportion   Solve the following problems....

Confidence Interval for Proportion

Solve the following problems.

1.  A survey of 200 workers showed 168 were interrupted 3 or more times an hours by phone messages. Find 90% confidence interval of the population proportion.

2. In a Gallup poll of 1005 individuals, 452 individuals thought they were worse off than a year ago. Find 95% confidence interval of the population proportion. 

Confidence Interval for Mean (Small Sample)

Solve the following problems 

1. A study of 28 employees of a company showed the mean distance they traveled to work was 14.3 miles and the sample standard deviation was 2 miles. Find the 95% confidence interval of the true mean.

2.  A sample of 6 college wrestlers had an average weight of 276 pounds with a sample standard deviation of 12 pounds. Find the 90% confidence interval for the population mean.

Answer & Explanation

The 90% confidence interval for the population proportion is (0.7974, 0.8826)

The 90% confidence interval for the population proportion is (0.4190, 0.4806)

The 95% confidence interval for the population mean is (13.5245, 15.0755)

The 90% confidence interval for the population mean is (266.1286, 285.8714)

Step 1: 1) Given that 

p^ = 168/200 = 0.84 

The 90% confidence level of z-critical = Z(0.10/2) = Z(0.05) = -+1.645                      (Use excel =NORM.S.INV(0.05))

Construction of 90% confidence interval for the population proportion is 

= 0.84 -+ (1.645)*sqrt((0.84)*(1-0.84)/200))

= 0.84 -+ 0.0426 = (0.7974, 0.8826)

Step 2: 2) Given that 

p^ = x/n = 452/1005 = 0.4498 

The 95% confidence level of Z-critical = Z(0.05/2) = Z(0.025) = -+1.960                 (Use excel =NORM.S.INV(0.025))

= 0.4498 -+ (1.960)*sqrt((0.4498)*(1-0.4498)/1005))

= 0.4498 -+ 0.0308 = (0.4190, 0.4806)

Step 3: 1 ) Given that 

Sample mean = xbar = 14.3

Sample sd = s = 2

t-critical = t(0.05/2, 28-1) = t(0.025, 27) = -+2.0518                                  (Use excel =T.INV.2T(0.05,27))

Construction of 95% confidence interval for the population mean is 

= 14.3 -+ (2.0518)*(2/sqrt(28))

= 14.3 -+ 0.7755 = (13.5245, 15.0755)

Step 4: 2) Given that 

Sample mean = xbar = 276

Sample sd = s = 12

t-critical = t(0.10/2, 6-1) = t(0.05, 5) = -+2.0150                                (Use excel =T.INV.2T(0.1,5))

Construction of 90% confidence interval for the population mean is 

= 276 -+ (2.0150)*(12/sqrt(6))

= 276 -+ 9.8714 = (266.1286, 285.8714)

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October 31, 2023

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The math problem that took nearly a century to solve

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What was Ramsey's problem, anyway?

A good problem fights back.

Journal information: arXiv

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Mathematics > Statistics Theory

Title: minimizing convex functionals over space of probability measures via kl divergence gradient flow.

Abstract: Motivated by the computation of the non-parametric maximum likelihood estimator (NPMLE) and the Bayesian posterior in statistics, this paper explores the problem of convex optimization over the space of all probability distributions. We introduce an implicit scheme, called the implicit KL proximal descent (IKLPD) algorithm, for discretizing a continuous-time gradient flow relative to the Kullback-Leibler divergence for minimizing a convex target functional. We show that IKLPD converges to a global optimum at a polynomial rate from any initialization; moreover, if the objective functional is strongly convex relative to the KL divergence, for example, when the target functional itself is a KL divergence as in the context of Bayesian posterior computation, IKLPD exhibits globally exponential convergence. Computationally, we propose a numerical method based on normalizing flow to realize IKLPD. Conversely, our numerical method can also be viewed as a new approach that sequentially trains a normalizing flow for minimizing a convex functional with a strong theoretical guarantee.

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