Ch. 5 - Discrete Probability Distributions

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 5 - Discrete Probability Distributions

Problem 5.2.2

Chapter 5, Problem 5.2.2

Notation Assume that we want to find the probability that when five speaking characters in movies are randomly selected, exactly two of them are females. Also assume that when randomly selecting a speaking character in a movie, the probability of getting a female is 0.331. Identify the values of n, x, p, and q.

Verified step by step guidance

Step 1: Understand the problem. This is a binomial probability problem where we are calculating the probability of exactly two successes (female speaking characters) in five trials (randomly selected speaking characters). The binomial distribution is characterized by the number of trials (n), the number of successes (x), the probability of success (p), and the probability of failure (q).

Step 2: Identify the number of trials (n). In this problem, five speaking characters are randomly selected, so n = 5.

Step 3: Identify the number of successes (x). The problem specifies that we are interested in exactly two females, so x = 2.

Step 4: Identify the probability of success (p). The probability of randomly selecting a female speaking character is given as 0.331, so p = 0.331.

Step 5: Identify the probability of failure (q). The probability of failure is the complement of the probability of success, calculated as q = 1 - p. Since p = 0.331, q = 1 - 0.331 = 0.669.

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Key Concepts

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

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. In this context, selecting a female character is considered a 'success,' and the distribution helps calculate the probability of obtaining a specific number of successes (e.g., exactly two females) in a given number of trials (e.g., five characters).

Parameters of the Binomial Distribution

In a binomial distribution, 'n' represents the total number of trials, 'x' is the number of successes we are interested in, 'p' is the probability of success on each trial, and 'q' is the probability of failure (1 - p). For this question, n is 5 (total characters), x is 2 (females), p is 0.331 (probability of selecting a female), and q is 0.669 (probability of selecting a male).

Probability Calculation

To find the probability of exactly x successes in n trials, we use the binomial probability formula: P(X = x) = (n choose x) * p^x * q^(n-x). This formula combines the number of ways to choose x successes from n trials with the probabilities of those successes and failures, allowing us to compute the likelihood of selecting exactly two female characters from five.

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