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Math Help - Probability Question

  1. #1
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    Probability Question

    Letís say that there is one lottery draw a week and if I was to play just one line per week until I die then I would have 1040 tries at winning the lottery.

    I read somewhere that I would have more chance of winning the lottery if I was to play all 1040 lines in one week and then never play again, as opposed to just playing one line a week for the next 20 years. (I.e. 1040 weeks.)

    Aren't the chances of me winning the lottery same in each case? (Albeit very slim!)
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  2. #2
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    Lets say the odds of winning the lotto are 1 in a million.

    The odds of winning if you play 1040 numbers in one week are 1040/Million = .00104

    The odds of winning if you play once a week for 1040 weeks are 1 - the odds of not winning = 1 - (999999/1000000)^1040 = .001039

    I believe however that the expected value of your lotto tickets are equal. The slightly lower probability of winning if you play once a week is made up by the slim chance that you hit the lotto more than once during that period.
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  3. #3
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    Quote Originally Posted by Penguins View Post
    Lets say the odds of winning the lotto are 1 in a million.

    The odds of winning if you play 1040 numbers in one week are 1040/Million = .00104

    The odds of winning if you play once a week for 1040 weeks are 1 - the odds of not winning = 1 - (999999/1000000)^1040 = .001039

    I believe however that the expected value of your lotto tickets are equal. The slightly lower probability of winning if you play once a week is made up by the slim chance that you hit the lotto more than once during that period.
    Thanks for that. I'm sure you are correct but I still don't quite get why the odds aren't exactly the same.

    I would have thought that either way I have 1040 tries to match a winning lottery number and the odds of each try are exactly the same.
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  4. #4
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    Imagine a game where I pick a number from 1 to 10 and you get to buy a number. If the number you buy is the number I chose, you win.

    Now you could pick all 10 numbers in 1 game and you are guaranteed to win.

    Or you could pick 1 number and play the game 10 times. Can you see how you would not be guaranteed to win this way?

    Again, the expected value would be the same in both cases, because you could end up winning more than once.


    Likewise for the 1 in a million lotto. Instead of playing 1040 times, you could play it a million times. So choose all 1 million number combinations and play once you are guaranteed to win. But if you play 1 number a million times, you might not. But again, you could win more than once.
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  5. #5
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    Quote Originally Posted by Penguins View Post
    Imagine a game where I pick a number from 1 to 10 and you get to buy a number. If the number you buy is the number I chose, you win.

    Now you could pick all 10 numbers in 1 game and you are guaranteed to win.

    Or you could pick 1 number and play the game 10 times. Can you see how you would not be guaranteed to win this way?
    I see what you mean. But if I did play the game 10 times, picking only one number, over and over again then on average I should win 1 in 10.

    Quote Originally Posted by Penguins View Post
    Again, the expected value would be the same in both cases, because you could end up winning more than once.
    So if it costs £1 a go and I got £10 back for a win then no matter whch method I chose, I would always break even. (Eventually!)

    EDIT: Sorry, I just had another thought. Taking the lottery as an example, if there is a 1 in a 14,000,000 chance of winning and I play just one line 14,000,000 times, what is the probability that I will win?
    Last edited by rede96; October 15th 2009 at 11:27 AM.
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  6. #6
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    Quote Originally Posted by rede96 View Post
    I see what you mean. But if I did play the game 10 times, picking only one number, over and over again then on average I should win 1 in 10.



    So if it costs £1 a go and I got £10 back for a win then no matter whch method I chose, I would always break even. (Eventually!)

    EDIT: Sorry, I just had another thought. Taking the lottery as an example, if there is a 1 in a 14,000,000 chance of winning and I play just one line 14,000,000 times, what is the probability that I will win?

    We'd solve that the same as in the first reply playing the lotto 1040 times.

    Probability of winning = 1 - probability of not winning = 1 - (13,999,999/14,000,000)^14,000,000 = 63.21%

    Again, the 63.21% includes the chances of winning more than once.
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  7. #7
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    I see, thanks for that Penguins.
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