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Math Help - Overlapping lottery chances of winning

  1. #1
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    Overlapping lottery chances of winning

    Can you please help me solve this probability problem?

    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
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  2. #2
    MHF Contributor
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    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?
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  3. #3
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    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
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