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Ch. 5 - Discrete Probability Distributions
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 5 - Discrete Probability DistributionsProblem 5.1.19b
Chapter 5, Problem 5.1.19b

Lottery. In Exercises 15–20, refer to the accompanying table, which describes probabilities for the California Daily 4 lottery. The player selects four digits with repetition allowed, and the random variable x is the number of digits that match those in the same order that they are drawn (for a “straight” bet).
Table showing probabilities for matching digits in a lottery: 0.656 for 0, 0.292 for 1, 0.049 for 2, 0.004 for 3, 0+ for 4.


Using Probabilities for Significant Events


b. Find the probability of getting 3 or more matches.

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1
Step 1: Understand the problem. The goal is to find the probability of getting 3 or more matches in the California Daily 4 lottery. This means we need to calculate the combined probability for x = 3 and x = 4.
Step 2: Refer to the provided table. The table lists the probabilities for different numbers of matching digits (x). Specifically, P(x=3) = 0.004 and P(x=4) = 0+.
Step 3: Add the probabilities for x = 3 and x = 4. Since the probability of x = 4 is given as 0+, it is effectively treated as 0 for practical purposes. Therefore, the total probability is P(x=3) + P(x=4).
Step 4: Write the formula for the calculation: \( P(x \geq 3) = P(x=3) + P(x=4) \). Substitute the values from the table into the formula.
Step 5: Interpret the result. The calculated probability represents the likelihood of getting 3 or more matches in the lottery. Ensure the addition is performed correctly to arrive at the final probability.

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Key Concepts

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

Probability Distribution

A probability distribution describes how the probabilities of a random variable are distributed across its possible values. In this case, the random variable x represents the number of matching digits in the lottery, and the table provides the probabilities for each possible outcome (0 to 4 matches). Understanding this distribution is essential for calculating the likelihood of specific events occurring.
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Calculating Probabilities in a Binomial Distribution

Cumulative Probability

Cumulative probability refers to the probability of a random variable being less than or equal to a certain value. To find the probability of getting 3 or more matches, one must calculate the cumulative probability for 3 and 4 matches. This involves summing the probabilities of these outcomes, which allows for a comprehensive understanding of the likelihood of achieving a certain level of success in the lottery.
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Complement Rule

The complement rule in probability states that the probability of an event occurring is equal to 1 minus the probability of it not occurring. In this context, to find the probability of getting 3 or more matches, one could alternatively calculate the probability of getting fewer than 3 matches (0, 1, or 2 matches) and subtract that from 1. This approach can simplify calculations and provide a clearer perspective on the desired outcome.
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Related Practice
Textbook Question

Politics The County Clerk in Essex, New Jersey, was accused of cheating by not using randomness in assigning the order in which candidates’ names appeared on voting ballots. Among 41 different ballots, Democrats were assigned the desirable first line 40 times. Assume that Democrats and Republicans are assigned the first line using a method of random selection so that they are equally likely to get that first line.


b. Find the probability of exactly 40 first lines for Democrats.

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Textbook Question

Binomial Probability Formula. In Exercises 13 and 14, answer the questions designed to help understand the rationale for the binomial probability formula.


Guessing Answers Standard tests, such as the SAT, ACT, or Medical College Admission Test (MCAT), typically use multiple choice questions, each with five possible answers (a, b, c, d, e), one of which is correct. Assume that you guess the answers to the first three questions.


c. Based on the preceding results, what is the probability of getting exactly one correct answer when three guesses are made?

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Textbook Question

Expected Value for the Florida Pick 3 Lottery In the Florida Pick 3 lottery, you can bet \$1 by selecting three digits, each between 0 and 9 inclusive. If the same three numbers are drawn in the same order, you win and collect \$500.


b. What is the probability of winning?


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Textbook Question

Lottery. In Exercises 15–20, refer to the accompanying table, which describes probabilities for the California Daily 4 lottery. The player selects four digits with repetition allowed, and the random variable x is the number of digits that match those in the same order that they are drawn (for a “straight” bet).


Using Probabilities for Significant Events


b. Find the probability of getting 2 or more matches.


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Textbook Question

In Exercises 25–28, find the probabilities and answer the questions.



Too Young to Tat Based on a Harris poll, among adults who regret getting tattoos, 20% say that they were too young when they got their tattoos. Assume that five adults who regret getting tattoos are randomly selected, and find the indicated probability.


b. Find the probability that exactly one of the selected adults says that he or she was too young to get tattoos.


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Textbook Question

Using Probabilities for Significant Events


b. Find the probability of getting 1 or fewer matches.

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