Final answer:
The probability P(5), in a geometric distribution with a probability of success p = 0.90, is calculated using the formula (1-p)^(k-1)p, resulting in 0.00009 or 0.009% when rounded to five decimal places.
Step-by-step explanation:
The student's question pertains to the geometric distribution, which is used to model the number of trials until the first success in a series of Bernoulli trials (each with the same probability of success p). We're given that the probability of success on any given trial is p = 0.90. The question asks to find P(5), which is the probability that the first failure occurs on the fifth trial.
To find P(5), we use the formula for the geometric probability, which is (1-p)^(k-1)p, where k is the trial number of the first success. Here, k = 5 because we are looking for the first failure to occur on the fifth trial. Applying the formula, we have:
P(5) = (1-0.90)^(5-1)×0.90 = (0.10)^4×0.90 = 0.0001×0.90 = 0.00009
Therefore, the probability P(5) when p=0.90 is 0.00009 or 0.009% when rounded to five decimal places.