My Math Forum Probability of Winning

 March 16th, 2007, 05:10 PM #1 Newbie   Joined: Mar 2007 Posts: 1 Thanks: 0 Probability of Winning Hello, OK, there is this home and car lottery in my area. They state that the odds of winning are 1 in 15. I assume that all they are doing is using the total number of prizes over total number of tickets to generate this result. For example 1,000 prizes to be won with 15,000 tickets being sold. This would equal a 1 in 15 chance of winning. Now my question is if I buy 3 tickets what are my odds and probability of winning. I thought that I could just use 1/15 + 1/15 + 1/15 = 3/15 or 1 in 5 chance. But then by that logic if I bought 15 tickets I should have odds of 1:1 which I know is not possible. What am I doing wrong????
 March 16th, 2007, 05:24 PM #2 Senior Member   Joined: Dec 2006 Posts: 1,111 Thanks: 0 That's right; If you buy 15 tickets, you are theoretically guaranteed to win. However, since theory isn't always reality, it's possible that by some accident you still won't get any prize, and somebody else who bought 15 tickets will wind up with two prizes. But, yes, you do theoretically have a 1:1 chance of winning if you buy 15 tickets. Of course, you still usually end up losing, because probability games are based on the idea that, say, each ticket costs $1, and the chance of winning is 1 in 15, but the prize is$10. Thus, even if you win lots of prizes, you still loose. It's just another example of the economic law of TANSTAAFL: There Ain't No Such Thing As A Free Lunch.
 March 16th, 2007, 06:15 PM #3 Newbie   Joined: Mar 2007 From: UK Posts: 21 Thanks: 0 If the chances of a ticket winning are 1/15, then the chances of it not winning are 14/15 Assuming that more one ticket can win at the same time (if there are 1000 prizes this should be possible), first calculate the probability that none of your tickets wins. This is (14/15)(14/15)(14/15) = 2744/3375. The probability that at least one of your tickets wins is therefore 1 - 2744/3375 = 631/3375. This is not equal to 1/15 + 1/15 + 1/15. If you buy 15 tickets, the chances of winning (using the above method) turn out to be 1 - (14/15)^15 = approximately 0.645. And this is not equal to 1. :P (However it is higher than 631/3375, which is approximately 0.187. Therefore buying more tickets does give you more chances of winning, which goes well with common sense. )
 March 17th, 2007, 05:18 AM #4 Senior Member   Joined: Dec 2006 Posts: 1,111 Thanks: 0 OK, but if we simplify the game using the same ratio, this time saying that there are only 15 tickets and 1 prize, then what you're saying is that buying 15 tickets would only give me a chance of winning of 0.645. It's pretty easy to see that that's false, since I am actually guaranteed to win. Of course, with more tickets and prizes, you are no longer guaranteed to win, but the probability of winning remains the same for a proportionate number of tickets. The probability of not winning is 1-(number of tickets bought)/15. Thus, if we buy 15 tickets, we have a 1 - 1 = 0 probability of not winning.
March 17th, 2007, 07:53 AM   #5
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Quote:
 Originally Posted by Infinity Of course, with more tickets and prizes, you are no longer guaranteed to win, but the probability of winning remains the same for a proportionate number of tickets.
No, the probability does not remain the same. It increases from 0.645 to 1. However, the expected number of winning tickets is the same.

March 17th, 2007, 11:44 AM   #7
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Quote:
 To guarantee at least one win you must buy at least 14,001 tickets.
...Yea, so I was participating in these sweepstakes in which there where 15,000 tickets and 1000 prizes, so I bought 14,001 tickets and won only 1 prize. I tell you, my luck must be runnin' out...

By the way, what are the odds of that actually occuring?

March 17th, 2007, 11:54 AM   #8
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 By the way, what are the odds of that actually occuring?
Would !15,000 / (!14,000*!1000) be correct?

March 17th, 2007, 04:10 PM   #9
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Quote:
Originally Posted by Infinity
Quote:
 To guarantee at least one win you must buy at least 14,001 tickets.
...Yea, so I was participating in these sweepstakes in which there where 15,000 tickets and 1000 prizes, so I bought 14,001 tickets and won only 1 prize. I tell you, my luck must be runnin' out...

By the way, what are the odds of that actually occuring?
14000!/15000^14000 is the chance of only your last ticket winning a prize; the odds are about 1 in 1.4e6497.

Otherwise, there are 14000 places the single winning ticket could go, so there are an additional 14000!/(15000^13999 * 1!) * 14000 possibilities, for a total chance of about 1 in 6.5e6488.

 March 17th, 2007, 04:16 PM #10 Senior Member   Joined: Nov 2006 From: I'm a figment of my own imagination :? Posts: 848 Thanks: 0 In other words, in order for only one of your tickets to win, you would have to be the most unlucky person in the universe .

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