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Ch. 5 - Discrete Probability Distributions
Triola - Elementary Statistics 14th Edition
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Tutti i libri di testoTriola 14th EditionCh. 5 - Discrete Probability DistributionsProblema 5.1.19a
Capitolo 5, Problema 5.1.19a

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


a. Find the probability of getting exactly 3 matches.

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Step 1: Understand the problem. The goal is to find the probability of getting exactly 3 matches in the California Daily 4 lottery. The table provided lists the probabilities for different numbers of matching digits (x).
Step 2: Locate the relevant probability in the table. The table shows the probability P(x) for each number of matching digits. For x = 3 (exactly 3 matches), the corresponding probability is 0.004.
Step 3: Interpret the probability. The value 0.004 represents the likelihood of selecting 3 digits that match the drawn digits in the same order during a 'straight' bet.
Step 4: Verify the context. Ensure that the table and problem description align with the calculation. The table explicitly provides the probability for x = 3, so no further computation is needed.
Step 5: Conclude the process. The probability of getting exactly 3 matches is directly obtained from the table as 0.004. This concludes the solution process.

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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 crucial for calculating the likelihood of specific events, such as getting exactly 3 matches.
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Calculating Probabilities in a Binomial Distribution

Random Variable

A random variable is a numerical outcome of a random phenomenon. In the context of the lottery question, the random variable x indicates the number of digits that match the drawn numbers in the same order. Recognizing how random variables function helps in analyzing the probabilities associated with different outcomes in probabilistic scenarios.
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Intro to Random Variables & Probability Distributions

Calculating Probabilities

Calculating probabilities involves determining the likelihood of a specific event occurring based on a probability distribution. For this question, to find the probability of getting exactly 3 matches, one would refer to the provided table and identify the corresponding probability value, which is 0.004. This process is fundamental in statistics for making informed predictions about random events.
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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.


a. Find the probability that none of the selected adults say that they were too young to get tattoos.


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In Exercises 9–16, use the Poisson distribution to find the indicated probabilities.


Births In a recent year (365 days), NYU-Langone Medical Center had 5942 births.


a. Find the mean number of births per day.

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In Exercises 5–8, assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 5.5 per year, as in Example 1; and proceed to find the indicated probability.

a. Find the probability that in a year, there will be 10 hurricanes.

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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.


a. Use the multiplication rule to find the probability that the first two guesses are wrong and the third is correct. That is, find P(WWC), where W denotes a wrong answer and C denotes a correct answer.

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In Exercises 29 and 30, assume that different groups of couples use the XSORT method of gender selection and each couple gives birth to one baby. The XSORT method is designed to increase the likelihood that a baby will be a girl, but assume that the method has no effect, so the probability of a girl is 0.5.


Gender Selection Assume that the groups consist of 36 couples.


a. Find the mean and standard deviation for the numbers of girls in groups of 36 births.


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In Exercises 5–8, assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 5.5 per year, as in Example 1; and proceed to find the indicated probability.

a. Find the probability that in a year, there will be no hurricanes.

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