Question

A face wash advertises that 85% of users will have clearer skin after using the wash for two weeks.

(a) Suppose 10 people use the wash according to the instructions. What is the probability that exactly 8 of them have clearer skin after two weeks?

b) Suppose 10 people use the wash according to the instructions. What is the probability that at least 8 of them have clearer skin after two weeks?

Answer 1

Answer 2

To solve both parts of the question, we will use the binomial probability formula:

P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)

where: P(X = k) is the probability of getting exactly k successes, n is the number of trials (in this case, the number of people using the wash), k is the number of successes we want (clearer skin), p is the probability of success on a single trial (probability of having clearer skin), and C(n, k) is the number of combinations of n things taken k at a time.

(a) Probability of exactly 8 people having clearer skin after two weeks:

n = 10 (number of people using the wash) k = 8 (number of people having clearer skin) p = 0.85 (probability of having clearer skin according to the advertisement)

P(X = 8) = C(10, 8) * 0.85^8 * (1 - 0.85)^(10 - 8)

C(10, 8) = 10! / (8! * (10-8)!) = 45

P(X = 8) = 45 * 0.85^8 * 0.15^2 P(X = 8) ≈ 0.3118 (rounded to four decimal places)

So, the probability that exactly 8 out of 10 people will have clearer skin after two weeks is approximately 0.3118.

(b) Probability of at least 8 people having clearer skin after two weeks:

To find this probability, we need to sum up the probabilities of getting 8, 9, or 10 successes:

P(X ≥ 8) = P(X = 8) + P(X = 9) + P(X = 10)

We have already calculated P(X = 8) in part (a).

P(X = 9) = C(10, 9) * 0.85^9 * 0.15^1 C(10, 9) = 10! / (9! * (10-9)!) = 10 P(X = 9) = 10 * 0.85^9 * 0.15^1 ≈ 0.3677

P(X = 10) = C(10, 10) * 0.85^10 * 0.15^0 C(10, 10) = 1 P(X = 10) = 1 * 0.85^10 * 0.15^0 ≈ 0.3277

P(X ≥ 8) ≈ 0.3118 + 0.3677 + 0.3277 ≈ 1.0072

Since probabilities cannot exceed 1, the final answer is 1 (100%).

So, the probability that at least 8 out of 10 people will have clearer skin after two weeks is 100%.