## Probability of Winning

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### 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????
wkunaman
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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.
Those who would sacrifice liberty to gain a little temporary safety, deserve neither liberty nor safety. -Benjamin Franklin
Infinity
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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. (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. )

JaneFairfax
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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.
Those who would sacrifice liberty to gain a little temporary safety, deserve neither liberty nor safety. -Benjamin Franklin
Infinity
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Infinity wrote: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.

trichoplax
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Let's say that exactly 15,000 tickets will be sold* and that 1,000 distinct tickets will get prizes. Then buying one ticket gives a you the advertised 1/15 chance of winning.

If you buy 15, your expected number of prizes is 1. Your odds of winning at least one prize is 1-(14/15)^15, or about 64.47%. This is made up for by your roughly 26.41% chance of winning more than once.

To guarantee at least one win you must buy at least 14,001 tickets.

* That is, if you buy more tickets someone else buys less to keep the total the same.

CRGreathouse
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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?
Those who would sacrifice liberty to gain a little temporary safety, deserve neither liberty nor safety. -Benjamin Franklin
Infinity
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By the way, what are the odds of that actually occuring?

Would !15,000 / (!14,000*!1000) be correct?
Those who would sacrifice liberty to gain a little temporary safety, deserve neither liberty nor safety. -Benjamin Franklin
Infinity
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Infinity wrote:
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.

CRGreathouse
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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 .
Robert Frost wrote:Two roads diverged in a wood, and I-
I took the one less travelled by,
And that has made all the difference.

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roadnottaken wrote: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 .

It's similar to the chance that you and I both happen to pick the same 900 people randomly from all people in the world.

CRGreathouse
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Infinity wrote: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.

You are now describing a totally new situation and probabilities have now changed. You can't use the probabilities calculated in the previous situation for the new situation â€“ these must be re-calculated.

JaneFairfax
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