Question

Pete is doing a science-experiment and have decided to work on it until it succeeds. The chance of success on any given day is 0.001. Let X be the number of days until he succeeds. Which probability distribution does X have? What is E(X)? What is the probabilty of Pete succeeding in his first year? If he doesn't succeed the first year, what's the probability of success in the second year?

Answer 1

Consider following Definition

“Assume Bernoulli trials — that is, (1) there are two possible outcomes, (2) the trials are independent, and (3) p, the probability of success, remains the same from trial to trial. Let X denote the number of trials until the first success. Then, the probability mass function of X is:

P(X=x)=(1−p)^x-1 p

for x = 1, 2, ... In this case, we say that X follows a geometric distribution with parameter p”.

For X ~ geometric (p), we know E(X) = 1/p

Assuming experiment conducted each day are independent of experiment conducted earlier days, then clearly X has geometric distribution with parameter p= 0.001.

Next, E(X) = 1/0.001 = 1000.

Next, probability of success in first year = P(X ≤ 365) = P(X= 0)+ P(X= 1)+……+ P(X= 365) = 0.3059 , (using calculator)

Next probability of success in second year if does not success in first year,

P(366 ≤ X ≤ 730 )= P(X = 366)+ P(X = 367)+………+ P(X = 730) = P(X ≤ 730) - P(X ≤ 366)

= 0.5183 – 0.3059 = 0.2124.