Table of contents

1. Intro to Stats and Collecting Data1h 14m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically2h 5m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables3h 6m
6. Normal Distribution and Continuous Random Variables2h 11m
7. Sampling Distributions & Confidence Intervals: Mean3h 23m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample3h 29m
10. Hypothesis Testing for Two Samples4h 50m
11. Correlation1h 6m
12. Regression1h 50m
13. Chi-Square Tests & Goodness of Fit1h 57m
14. ANOVA1h 57m

5. Binomial Distribution & Discrete Random Variables

Binomial Distribution

Statistics

5. Binomial Distribution & Discrete Random Variables

Binomial Distribution

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2:04 minutes

Problem 5.CRE.2b

Textbook Question

Kentucky Pick 4 In the Kentucky Pick 4 lottery game, you can pay $1 for a “straight” bet in which you select four digits with repetition allowed. If you buy only one ticket and win, your prize is $2500.

b. If you play this game once every day, find the mean number of wins in years with exactly 365 days.

Verified step by step guidance

Step 1: Understand the problem. The Kentucky Pick 4 lottery involves selecting four digits with repetition allowed, meaning each digit can range from 0 to 9. The total number of possible combinations is calculated as 10^4, since there are 10 choices for each of the 4 digits.

Step 2: Calculate the probability of winning. Since there is only one winning combination out of the total possible combinations, the probability of winning on a single ticket is given by P(win) = 1 / (10^4).

Step 3: Determine the number of days played in a year. The problem states that the game is played once every day in a year with 365 days. Therefore, the total number of games played in a year is 365.

Step 4: Use the formula for the mean number of wins. The mean number of wins in a given number of trials is calculated as μ = n * P(win), where n is the number of trials (365 days) and P(win) is the probability of winning on a single ticket.

Step 5: Substitute the values into the formula. Replace n with 365 and P(win) with 1 / (10^4) to compute the mean number of wins. The result will represent the expected number of wins in a year with 365 days.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In the context of the Kentucky Pick 4 lottery, the probability of winning with a single ticket can be calculated based on the total number of possible combinations of four digits, considering that repetition is allowed.

Expected Value

Expected value is a statistical concept that provides a measure of the center of a probability distribution, representing the average outcome if an experiment is repeated many times. In this lottery scenario, the expected number of wins over a year can be calculated by multiplying the probability of winning by the number of plays (365 days).

Mean

The mean, or average, is a fundamental statistical measure that summarizes a set of values by dividing the sum of those values by the number of values. In this case, the mean number of wins in a year can be derived from the expected value, indicating how many times a player can expect to win if they play the lottery daily.

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Textbook Question

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a. Find the probability that three random people killed by lightning strikes are all males.

Textbook Question

Kentucky Pick 4 In the Kentucky Pick 4 lottery game, you can pay $1 for a “straight” bet in which you select four digits with repetition allowed. If you buy only one ticket and win, your prize is $2500.

a. If you buy one ticket, what is the probability of winning?

Textbook Question

c. If you play this game once every day, find the probability of no wins in 365 days.

Textbook Question

Tennis Challenge In a recent U.S. Open tennis tournament, there were 945 challenges made by singles players, and 255 of them resulted in referee calls that were overturned. The accompanying table lists the results by gender.

a. If 1 of the 945 challenges is randomly selected, what is the probability that it resulted in an overturned call?

Textbook Question

Exercises 33 and 34 involve the method of composite sampling, whereby a medical testing laboratory saves time and money by combining blood samples for tests so that only one test is conducted for several people. A combined sample tests positive if at least one person has the disease. If a combined sample tests positive, then individual blood tests are used to identify the individual with the disease or disorder.

HIV It is estimated that in the United States, the proportion of people infected with the human immunodeficiency virus (HIV) is 0.00343. In tests for HIV, blood samples from 50 different people are combined. What is the probability that the combined sample tests positive for HIV? Is it unlikely for such a combined sample to test positive?

Textbook Question

b. If one of the overturned calls is randomly selected, what is the probability that the challenge was made by a woman?

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