On the package for a certain brand of okra seeds there is a guarantee that, if the printed instructions are followed, $50 \%$ of planted seeds will germinate. If this percentage is correct, what is the probability that, in a random sample of 7 seeds, exactly 3 germinate?
Round your answer to three decimal places.
Answer
Final Answer: The probability that, in a random sample of 7 seeds, exactly 3 germinate is \(\boxed{0.273}\)
Step 1 ：This problem is a binomial probability problem. The binomial distribution model is appropriate for a statistical experiment if the following conditions are met: 1. The experiment consists of n repeated trials. 2. Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure. 3. The probability of success, denoted by P, is the same on every trial. 4. The trials are independent; the outcome on one trial does not affect the outcome on other trials.
Step 2 ：In this case, we have n=7 trials (seeds), each of which can either germinate (success) or not germinate (failure). The probability of success is 0.5 (50%), and we assume that the outcome for each seed is independent of the others. We want to find the probability that exactly 3 seeds germinate, which is a typical binomial probability problem.
Step 3 ：The formula for binomial probability is: \(P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))\) where: - \(P(X=k)\) is the probability we want to find, - \(C(n, k)\) is the number of combinations of n items taken k at a time, - p is the probability of success on a single trial, - n is the number of trials, and - k is the number of successes we want.
Step 4 ：In this case, n=7, p=0.5, and k=3. We can plug these values into the formula to find the answer.
Step 5 ：First, calculate the number of combinations: \(C(n, k) = C(7, 3) = 35\)
Step 6 ：Next, calculate the probability of success on a single trial: \(p^k = 0.5^3 = 0.125\)
Step 7 ：Then, calculate the probability of failure on the remaining trials: \((1-p)^(n-k) = (1-0.5)^(7-3) = 0.0625\)
Step 8 ：Finally, multiply these values together to find the probability: \(P(X=3) = C(n, k) * (p^k) * ((1-p)^(n-k)) = 35 * 0.125 * 0.0625 = 0.273\)
Step 9 ：Final Answer: The probability that, in a random sample of 7 seeds, exactly 3 germinate is \(\boxed{0.273}\)
