Suppose you take a pass-fail test repeatedly. Let Sk be the event that you are successful in your kth try, and Fk be the event that you fail the test in your kth try. On your first try, you have a 50 percent chance of passing the test.

P(S1) = 1 − P(F1) = 1/2.

Assume that as you take the test more often, your chance of failing the test goes down. In particular,

P(Fk)=12⋅P(Fk−1), for k=2,3,4,⋯

However, the result of different exams are independent. Suppose you take the test repeatedly until you pass the test for the first time. Let X be the total number of tests you take, so Range(X)={1,2,3,⋯}.

Find P(X=1),P(X=2),P(X=3).

Find a general formula for P(X=k) for k=1,2,⋯.

Find the probability that you take the test more than 2 times.

Given that you take the test more than once, find the probability that you take the test exactly twice.