Question

The manufacturer of a fertilizer guarantees that with the aid of the fertilizer, 70% of planted seeds will germinate. Suppose the manufacturer is correct. If 10 seeds planted with the fertilizer are randomly selected, what is the probability that more than 4 of them germinate? Carry your intermediate computations to at least four decimal places, and round your answer to two decimal places.

Answer 1

The probability that more than 4 out of 10 seeds germinate is found using the binomial probability formula. The probabilities are calculated for 0 to 4 germinating seeds, summed, and then subtracted from 1 to find the probability of more than 4 germinating seeds.

Step-by-step explanation:

The probability that more than 4 out of 10 seeds will germinate, given that the manufacturer guarantees a germination rate of 70%, can be calculated using the binomial probability formula:

P(X > 4) = 1 - (P(X ≤ 4))

To find P(X ≤ 4), we add the probabilities of exactly 0, 1, 2, 3, and 4 seeds germinating:

• P(X = 0) = (0.3)^10
• P(X = 1) = C(10,1) ⋅ (0.7)^1 ⋅ (0.3)^9
• P(X = 2) = C(10,2) ⋅ (0.7)^2 ⋅ (0.3)^8
• P(X = 3) = C(10,3) ⋅ (0.7)^3 ⋅ (0.3)^7
• P(X = 4) = C(10,4) ⋅ (0.7)^4 ⋅ (0.3)^6

After calculating each of these probabilities to at least four decimal places, we then sum them up to find P(X ≤ 4), and subtract that sum from 1 to find P(X > 4):

1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)]

For our final answer, we round to two decimal places as instructed.