Textbook Question
Find the mean of the random variable x described in the preceding exercise.
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Ch. 5 - Discrete Probability Distributions
Triola14th EditionElementary StatisticsISBN: 9780137366446
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Table of contents
- Ch. 1 - Introduction to Statistics88
- Ch. 2 - Exploring Data with Tables and Graphs70
- Ch. 3 - Describing, Exploring, and Comparing Data66
- Ch. 4 - Probability112
- Ch. 5 - Discrete Probability Distributions109
- Ch. 6 - Normal Probability Distributions122
- Ch. 7 - Estimating Parameters and Determining Sample Sizes107
- Ch. 8 - Hypothesis Testing75
- Ch. 9 - Inferences from Two Samples94
- Ch. 10 - Correlation and Regression80
- Ch. 11 - Goodness-of-Fit and Contingency Tables29
- Ch. 12 - Analysis of Variance42
Chapter 5, Problem 5.CRE.2a
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?
Verified step by step guidance1
Understand the problem: In the Kentucky Pick 4 lottery, you select four digits (0-9) with repetition allowed. The total number of possible outcomes is the total number of 4-digit combinations where each digit can range from 0 to 9.
Calculate the total number of possible outcomes: Since each digit has 10 possible values (0 through 9), and there are 4 digits, the total number of outcomes is given by \( 10^4 \).
Determine the number of favorable outcomes: To win, the exact 4-digit combination you selected must match the winning combination. Therefore, there is only 1 favorable outcome.
Calculate the probability of winning: Probability is defined as the ratio of favorable outcomes to total outcomes. Use the formula \( P(\text{winning}) = \frac{\text{favorable outcomes}}{\text{total outcomes}} \). Substitute the values from the previous steps.
Simplify the probability expression: Simplify the fraction \( \frac{1}{10^4} \) to express the probability in its simplest form.
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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 by determining the total number of possible outcomes and the number of favorable outcomes (winning combinations).
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Introduction to Probability
Combinatorics
Combinatorics is a branch of mathematics dealing with combinations and permutations of objects. In the Kentucky Pick 4 game, since digits can be repeated, the total number of possible combinations can be calculated using the formula for permutations with repetition, which is essential for determining the total outcomes in the lottery.
Expected Value
Expected value is a key concept in statistics that represents the average outcome of a random event when considering all possible outcomes and their probabilities. In the lottery context, understanding the expected value can help players assess whether the potential prize justifies the cost of the ticket, given the low probability of winning.
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04:14Expected Value (Mean) of Random Variables
Related Practice
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.
c. If you play this game once every day, find the probability of no wins in 365 days.
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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.
c. If two different challenges are randomly selected without replacement, find the probability that they both resulted in an overturned call.
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Textbook Question
Is the mean found in the preceding exercise a statistic or a parameter?
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.
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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Textbook Question
In Exercises 6–10, refer to the accompanying table, which describes the numbers of adults in groups of five who reported sleepwalking (based on data from “Prevalence and Comorbidity of Nocturnal Wandering In the U.S. Adult General Population,” by Ohayon et al., Neurology, Vol. 78, No. 20).
Significant Events Is 4 a significantly high number of sleepwalkers in a group of 5 adults? Explain.
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