In Exercises 13–16, find the indicated binomial probabilities. If convenient, use technology or Table 2 in Appendix B.
Fifty-three percent of U.S. adults support attempting to land an astronaut on Mars. You randomly select eight U.S. adults. Find the probability that the number who support attempting to land an astronaut on Mars is (b) at least three

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 = 8 (number of trials), p = 0.53 (probability of success), and we are looking for the probability of at least 3 successes.

Step 2: Define the complement event. The probability of 'at least 3 successes' can be calculated as 1 minus the probability of 'fewer than 3 successes.' Mathematically, this is expressed as P(X ≥ 3) = 1 - P(X < 3).

Step 3: Break down P(X < 3). This represents the probability of having 0, 1, or 2 successes. Using the binomial probability formula, P(X = k) = (n choose k) * p^k * (1-p)^(n-k), calculate P(X = 0), P(X = 1), and P(X = 2).

Step 4: Add the probabilities for P(X = 0), P(X = 1), and P(X = 2). This gives P(X < 3). Use the binomial formula for each term: P(X = k) = (8 choose k) * (0.53)^k * (0.47)^(8-k), where k = 0, 1, 2.

Step 5: Subtract the result of P(X < 3) from 1 to find P(X ≥ 3). This final step gives the probability of at least 3 successes. If using technology or a binomial probability calculator, input n = 8, p = 0.53, and calculate P(X ≥ 3) directly.

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주요 개념

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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, the trials are the responses of the eight randomly selected U.S. adults regarding their support for landing an astronaut on Mars, with a success defined as a 'yes' response.

Probability Calculation

To find the probability of a specific number of successes in a binomial distribution, we use the binomial probability formula: 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. For this question, we need to calculate the probability of at least three supporters, which involves summing the probabilities of getting three, four, five, six, seven, or eight supporters.

Cumulative Probability

Cumulative probability refers to the probability of obtaining a value less than or equal to a certain number in a distribution. In this case, to find the probability of at least three supporters, we can either calculate the probabilities for three or more supporters directly or use the complement rule by finding the cumulative probability of having fewer than three supporters and subtracting it from one.