# Thread: Overlapping lottery chances of winning

1. ## Overlapping lottery chances of winning

There are two lotteries for admission into a popular concert. The concert venue holds 85,000 people, all of whom will be selected through two lotteries. The first lottery consists of 30,000 "pre-sale" ticket holders vying for 20,000 seats. The 10,000 pre-sale ticket holders that are not selected for admission in the first lottery are then rolled over to the general lottery.

The general lottery has 100,000 ticket holders (10,000 pre-sale losers + 90,000 normal ticket holders) vying for 65,000 seats.

What is the chance that a pre-sale ticket holder will make it into the concert?

I thought that I would be able to solve this by adding the two probabilities of winning and subtracting the product of the two probabilities of not winning, but when I do this, I get a probability over 100%. Like this:

20,000 / 30,000 = 66.6666% (Pre-sale lottery chance of winning) = 33.3334% chance of losing
65,000 / 100,000 = 65% (Normal lottery chance of winning) = 35% chance of losing
66.6666% + 65% = 131.6666% - (33.3334% * 35%) = 131.6666% - 11.6667% = 119.9999%

Clearly, the chances of a pre-sale lottery ticket holder winning a slot are not over 100%, so I must be doing something wrong. Can you help please?

Thank you!
-S

2. ## Re: Overlapping lottery chances of winning

Hey Sarfa33.

The first thing you should do is form a distribution for the number of pre-sale people and normal ticket holders getting into the lottery.

Have you heard of the Negative-Binomial distribution?

3. ## Re: Overlapping lottery chances of winning

Actually, I think I figured out how to do it.

Can I just multiply the chances of NOT winning each round of the lottery together to get the combined probability of losing?

So in my example:

20,000 / 30,000 = 66.6666% (Pre-sale lottery chance of winning) = 33.3334% chance of losing
65,000 / 100,000 = 65% (Normal lottery chance of winning) = 35% chance of losing

35% * 33.3334% = 11.6667% chance of losing both lotteries = 88.333% chance of getting into the concert

Is that right?

Thanks!
-S