Forty-nine percent of U.S. adults think that human activity such as burning fossil fuels contributes a great deal to climate change. You randomly select 25 U.S. adults. Find the probability that the number who think that human activity contributes a great deal to climate change is (a) exactly 12,

검증된 단계별 안내

Step 1: Recognize that this is a binomial probability problem. The binomial distribution is used when there are a fixed number of independent trials (n), each with two possible outcomes (success or failure), and the probability of success (p) is constant for each trial. Here, n = 25 (number of trials) and p = 0.49 (probability of success).

Step 2: Use the binomial probability formula to calculate the probability of exactly 12 successes. The formula is: P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where 'n choose k' is the binomial coefficient, p is the probability of success, and k is the number of successes. Here, k = 12.

Step 3: Calculate the binomial coefficient (n choose k), which is given by the formula: (n choose k) = n! / [k! * (n-k)!]. Substitute n = 25 and k = 12 into this formula.

Step 4: Substitute the values of p = 0.49, k = 12, and n = 25 into the binomial probability formula. Compute p^k (0.49^12) and (1-p)^(n-k) ((1-0.49)^(25-12)).

Step 5: Multiply the binomial coefficient from Step 3 by the probabilities calculated in Step 4 to find the final probability P(X = 12).

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주요 개념

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

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, each adult's opinion on climate change can be seen as a trial, where 'success' is defined as an adult believing that human activity contributes significantly to climate change. The distribution is characterized by two parameters: the number of trials (n) and the probability of success (p).

Probability Mass Function (PMF)

The probability mass function gives the probability of obtaining exactly k successes in n trials for a binomial distribution. It is calculated using the formula P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where 'n choose k' represents the binomial coefficient. This function is essential for determining the likelihood of observing a specific number of successes, such as exactly 12 adults in this scenario.

Normal Approximation to the Binomial

For large sample sizes, the binomial distribution can be approximated by a normal distribution, which simplifies calculations. This approximation is valid when both np and n(1-p) are greater than 5. In this case, with n = 25 and p = 0.49, the normal approximation can be used to estimate probabilities, making it easier to analyze the distribution of opinions among the selected adults.