Correct option is D
Concept:
Uniform Distribution: In a uniform distribution, all outcomes within a given range are equally likely. Here, the length
of the fish is uniformly distributed between 2 and 4 inches, meaning every length in that range has the same probability.
Solution:
The length of bristlemouth fish is uniformly distributed between 2 and 4 inches.
This means every length between 2 and 4 inches has an equal probability of being selected.
The total range of fish lengths is from 2 to 4 inches, so the range is 4 - 2 = 2 inches.
The range of fish lengths that are 3 inches or longer is from 3 to 4 inches. So the favorable range is 4 - 3 = 1 inch.
The probability of catching a fish that is 3 inches or longer is given by the ratio of the range to the total range:
P(Fish ≥ 3 inches) = 1/2 = 0.5.
The probability of not catching a fish that is 3 inches or longer (i.e., catching a fish shorter than 3 inches) is:
P(Fish < 3 inches) = 1 - 0.5 = 0.5.
If a fisherman catches 5 fish, the probability that none of the 5 fish are 3 inches or longer is:
P(None of the 5 fish ≥ 3 inches) = (0.5)^5 = 0.03125.
The probability that at least one fish is 3 inches or longer is the complement of the probability that none of the fish are 3 inches or longer:
P(At least one fish ≥ 3 inches) = 1 - 0.03125 = 0.96875.
The probability that at least one of the 5 bristlemouth fish is 3 inches or longer is 0.96875.
