A certain type of new business succeeds 60% of the time. Suppose that three such businesses open (where they do not compete with each other, so it is reasonable to believe that their relative successes would be independent). The probability that exactly one business succeeds is?
To find the probability that exactly one business succeeds, we need to consider the different ways in which one business can succeed and the other two fail.
Let's define the following events:
A: Business 1 succeeds, Business 2 fails, Business 3 fails
B: Business 1 fails, Business 2 succeeds, Business 3 fails
C: Business 1 fails, Business 2 fails, Business 3 succeeds
Since the businesses do not compete with each other and their relative successes are independent, the probability of each event can be calculated as follows:
P(A) = 0.6 * 0.4 * 0.4 = 0.096
P(B) = 0.4 * 0.6 * 0.4 = 0.096
P(C) = 0.4 * 0.4 * 0.6 = 0.096
The probability that exactly one business succeeds is the sum of the probabilities of events A, B, and C:
P(exactly one business succeeds) = P(A) + P(B) + P(C) = 0.096 + 0.096 + 0.096 = 0.288
Therefore, the probability that exactly one business succeeds is 0.288 or 28.8%.