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.

검증된 단계별 안내

Step 1: Recognize that this is a binomial probability problem. The problem involves a fixed number of trials (n = 5), two possible outcomes (success: the adult says they were too young, and failure: the adult does not say they were too young), and a constant probability of success (p = 0.20).

Step 2: Use the binomial probability formula to calculate the probability of exactly one success. The formula is: P(X = k) = (n choose k) * p^k * (1 - p)^(n - k), where 'n' is the number of trials, 'k' is the number of successes, and 'p' is the probability of success.

Step 3: Substitute the given values into the formula. Here, n = 5, k = 1, and p = 0.20. The formula becomes: P(X = 1) = (5 choose 1) * (0.20)^1 * (1 - 0.20)^(5 - 1).

Step 4: Calculate the binomial coefficient (5 choose 1), which is the number of ways to choose 1 success from 5 trials. This is given by the formula: (n choose k) = n! / [k! * (n - k)!]. For (5 choose 1), this simplifies to 5.

Step 5: Combine all the components to express the probability. Multiply the binomial coefficient (5), the probability of success raised to the power of k ((0.20)^1), and the probability of failure raised to the power of (n - k) ((0.80)^4). This gives the final expression for P(X = 1).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Binomial Probability

Binomial probability refers to the likelihood of a specific number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, the 'success' is defined as an adult stating they were too young to get a tattoo. The formula for calculating this probability is P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success.

Probability of Success and Failure

In probability theory, the probability of success is the chance of an event occurring, while the probability of failure is the chance of it not occurring. For this problem, the probability of success (an adult saying they were too young) is 20% (or 0.2), and the probability of failure (not saying they were too young) is 80% (or 0.8). Understanding these probabilities is crucial for calculating the overall binomial probability.

Combinatorial Coefficient

The combinatorial coefficient, often represented as 'n choose k' or C(n, k), calculates the number of ways to choose k successes from n trials without regard to the order of selection. This is essential in binomial probability calculations, as it accounts for the different combinations of successes and failures. For example, in this scenario, it helps determine how many ways one adult can be identified as saying they were too young among the five selected.