Question

In the past year, 15% of businesses have eliminated jobs. If 6 businesses are selected at random, find the probability that at least 4 have eliminated jobs during the last year.

Answer 1

To find the probability of at least 4 out of 6 businesses having eliminated jobs in the past year, we can use the binomial probability formula. Calculate the probabilities for each scenario and sum them up to get the final probability.

Step-by-step explanation:

To find the probability that at least 4 out of 6 randomly selected businesses have eliminated jobs in the past year, we need to consider the different possible outcomes. Let's break it down step by step:

1. First, let's calculate the probability that exactly 4 out of 6 businesses have eliminated jobs. We can use the binomial probability formula: P(X=k) = C(n, k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, C(n, k) is the number of combinations of n things taken k at a time, and p is the probability of success on a single trial.
2. For this question, n = 6, k = 4, and p = 0.15. Plugging these values into the formula, we get: P(X=4) = C(6, 4) * 0.15^4 * (1-0.15)^(6-4) = 15 * 0.15^4 * 0.85^2.
3. Next, let's calculate the probability that exactly 5 out of 6 businesses have eliminated jobs. Using the same formula as before, we get: P(X=5) = C(6, 5) * 0.15^5 * (1-0.15)^(6-5) = 6 * 0.15^5 * 0.85^1.
4. Finally, let's calculate the probability that all 6 out of 6 businesses have eliminated jobs. Using the formula, we get: P(X=6) = C(6, 6) * 0.15^6 * (1-0.15)^(6-6) = 0.15^6 * 0.85^0 = 0.15^6.
5. To find the probability that at least 4 out of 6 businesses have eliminated jobs, we need to add up the probabilities of these three scenarios: P(at least 4) = P(X=4) + P(X=5) + P(X=6).
6. Calculate the values for P(X=4), P(X=5), and P(X=6) and sum them up to get the final probability.