Ch. 5 - Discrete Probability Distributions

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 5 - Discrete Probability Distributions

Problem 5.2.38c

Chapter 5, Problem 5.2.38c

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.

c. Find the probability of 40 or more first lines for Democrats.

Verified step by step guidance

Step 1: Define the problem as a binomial probability scenario. Here, the number of trials (n) is 41 (the number of ballots), and the probability of success (p) for Democrats being assigned the first line is 0.5 (since the selection is random and equally likely).

Step 2: Identify the random variable X, which represents the number of times Democrats are assigned the first line. X follows a binomial distribution: X ~ Binomial(n=41, p=0.5).

Step 3: To find the probability of Democrats being assigned the first line 40 or more times, calculate P(X ≥ 40). This can be expressed as the sum of probabilities: P(X = 40) + P(X = 41).

Step 4: Use the binomial probability formula to calculate P(X = k), where k is the number of successes: P(X = k) = (n choose k) * p^k * (1-p)^(n-k). For P(X = 40), substitute k=40, n=41, and p=0.5 into the formula. Similarly, calculate P(X = 41).

Step 5: Add the probabilities P(X = 40) and P(X = 41) to find P(X ≥ 40). Alternatively, you can use statistical software or a calculator with binomial distribution functions to compute this probability efficiently.

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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 this context, it helps determine how likely it is for Democrats to be assigned the first line on the ballot 40 times out of 41, assuming a fair random selection process.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. Here, it applies to the scenario of assigning first lines to candidates, where each assignment can be seen as a trial with two outcomes: Democrat or Republican.

Normal Approximation

The normal approximation to the binomial distribution is used when the number of trials is large, allowing for easier calculations of probabilities. In this case, it can simplify finding the probability of Democrats receiving the first line 40 times by approximating the binomial distribution with a normal distribution.

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Using the Normal Distribution to Approximate Binomial Probabilities

Related Practice

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.

c. Find the probability that the number of selected adults saying they were too young is 0 or 1.

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Textbook Question

Using Probabilities for Significant Events

c. Which probability is relevant for determining whether 1 is a significantly low number of matches: the result from part (a) or part (b)?

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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.

d. If 1 of the 945 challenges is randomly selected, find the probability that it was made by a man or was upheld with an overturned call.

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Textbook Question

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.

c. Find the probability that in a single day, there are no births. Would 0 births in a single day be a significantly low number of births?

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Textbook Question

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.

c. Find the probability of 152 or more yellow peas.

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Textbook Question

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.

c. Is a result of 7 peas with green pods a result that is significantly low? Why or why not?

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