Probability Help

#1
I'm having some trouble trying to figure this out:

James is going to write an exam
Suppose that if he hadn't passed in previous attempts, each time he attempts the exam, he has probability p to pass . The probability that he passes on his fi rst try is 5 times that of passing on his second try.
1) Find E(X)
2) Find the probability that James takes more than 10 times to finish the exam.
3) From (2), deduce that for such distribution, for any n > 0:
P(X > n) = (1 - p)^n

I believe that I should be using binomial distribution in some way. I'm not exactly sure to find the PMF, which I think is required?

Any help is appreciated
 
#2
This is a Negative Binomial Distribution:

\(P_{r,p} = {x + r - 1 \choose{r - 1} }p^r (1-p)^x \)

r = number of successes
x = number of failures = 0, 1, 2 ...
x + r = number of trials

p = probability of success
q = probability of failure = 1 - p

When r = 1 this simplifies to

P(0) = p
P(1) = p (1 - p)

We are given P(0) = 5 * P(1)

p = 5 * p * (1 - p)

1/5 = 1 - p

p = 4/5

Hope this gets you started
 
#3
Thank you for the help, that gave me a lot to work with! If you don't mind me asking, what gave it away that it is a negative binomial distribution?
 
#4
Well Binomial distribution usually has fixed number of trials. Each event is independent and can only have one of two outcomes. Flip a coin 20 times. What is the probability of getting 10 heads, less than 7 heads, etc.

Negative Binomial Distribution is when you repeat the experiment an unknown number of times until you have a fixed number of successes. Or failures depending on your textbook. Take a test, repeat until you pass, flip a coin until you get heads. Or until you get 3 tails. Roll two dice until you get a 7, etc. Like binomial distribution, each trial is independent and can only have one of two outcomes.