You decide to play monthly in two different lotteries, and you stop playing as s

Question

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 p_1 for one of the lotteries and p_2 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? If p_1 = 2 times 10^-6 and p_2 = 3 times 10^-6 what is the probability of winning if you participate in both lotteries 100,000 times.

Explanation / Answer

Let S be the event that you stop playing the lottery in a given month

. According to the problem, S occurs if you either •

Win lottery 1 - we’ll call this event A • Win lottery 2 -

we’ll call this event B • Win both lotteries

- this would be event A B And there’s no other way to make you stop playing.

{Stop Playing} = {Win both lotteries} OR {Win 1 st, not 2 nd} OR {Win 2 nd, not 1 st}.

That is, S = (A B) (A B^ c ) (A ^c B).

Therefore P(S) = p1p2 + p1(1 p2) + (1 p1)p2

Thus M would have a geometric distribution with parameter P(S)

bans) p1=2*10^-6 p2 = 3*10^ -6

probability of winning = P(S) = p1p2 + p1(1 p2) + (1 p1)p2

= (2*10^-6 )3*10^ -6 +2*10^ -6(1-3*10^ -6)+(1-2*10^ -6) *3*10^ -6

  

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