Probability Mass Function For Winning the Lottery

[BlueScreenOD]

Homework Statement

You decide to play monthly in two different lotteries, and you stop playing
as soon as you win a prize in one (or both) lotteries of at least one million
euros. Suppose that every time you participate in these lotteries, the probability
to win one million (or more) euros is p1 for one of the lotteries and p2
for the other. Let M be the number of times you participate in these lotteries
until winning at least one prize. What kind of distribution does M have, and
what is its parameter?

Homework Equations

Binomial Distribution: bin(n, p)
px(k) = p(X = K) = (n choose k) * p^k * (1 -p)^(N-k) for k = 0, 1, ..., n

Gemetric distribution: Geo(p):
Px(k) = P(X = k) = (1-p)^(k-1)p for k = 1, 2, ...

The Attempt at a Solution

I'm pretty sure this is a geometric distribution.

However, I'm not quite sure what the p is. I think it's
p1 * (1 - p2) + p2 * (1 - p1) + p1 * p2 =
p1 + p2 - p1p2

so would the parameter be (p1 + p2 - p1p2)?

 

[Dick]

I think you are right. Your probability at stopping at each stage is the probability of winning lottery 1 OR lottery 2. As you say, that's p=p1+p2-p1*p2. So to reach the kth stage you have have not stopped k-1 times and stopped once. (1-p)^(k-1)*p.