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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 5.1.18a
Capitolo 5, Problema 5.1.18a

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 2 matches.

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Step 1: Understand the problem. The question asks for the probability of getting exactly 2 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. From the table, the probability of getting exactly 2 matches corresponds to P(x = 2).
Step 3: Identify the value of P(x = 2) from the table. According to the table, P(x = 2) is listed as 0.049.
Step 4: Interpret the result. This means that the probability of getting exactly 2 matches in the California Daily 4 lottery is 0.049, or 4.9%.
Step 5: Conclude the solution. The probability value is directly obtained from the table, and no further calculations are necessary for this problem.

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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 essential for calculating specific probabilities, such as the likelihood of getting exactly 2 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, the random variable x quantifies the number of digits that match the drawn numbers in the correct order. Recognizing how random variables operate helps in analyzing the probabilities associated with different outcomes, which is crucial for solving the given problem.
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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 2 matches, one would refer to the provided table and identify the corresponding probability value, which is 0.049. This process is fundamental in statistics for making informed predictions and decisions based on random events.
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Using Probabilities for Significant Events


a. Find the probability of getting exactly 1 match.

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In Exercises 31 and 32, assume that hybridization experiments are conducted with peas having the property that for offspring, there is a 0.75 probability that a pea has green pods (as in one of Mendel’s famous experiments).


Hybrids Assume that offspring peas are randomly selected in groups of 16.


a. Find the mean and standard deviation for the numbers of peas with green pods in the groups of 16.

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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.


Hurricanes


a. Find the probability that in a year, there will be 7 hurricanes.

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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.


a. Use the range rule of thumb to identify the limits separating values that are significantly low and those that are significantly high. Based on the results, is the result of 40 first lines for Democrats significantly high?

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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.

b. In a 118-year period, how many years are expected to have no hurricanes?

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One of Mendel’s famous experiments with peas resulted in 580 offspring, and 152 of them were yellow peas. Mendel claimed that under the same conditions, 25% of offspring peas would be yellow. Assume that Mendel’s claim of 25% is true, and assume that a sample consists of 580 offspring peas.


b. Find the probability of exactly 152 yellow peas.


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