Chances of Winning a RaffleDate: 15 Feb 1995 20:50:06 -0500 From: Anonymous Subject: Re: probability problem Dear Dr. Math: I have a problem with probability that I can't seem to solve. In fact, I don't know how to approach it. There is a concert and tickets are to be raffled off. There are 3500 winners and 175,000 applicants. I sent in 100 applications. What are my chances of winning? Looking forward to your answer. - Mike DiBianco Date: 15 Feb 1995 20:50:06 -0500 From: Anonymous Subject: Re: probability problem Mike -- It's easier first to compute your chances of NOT winning. Your first application will be a loser with probability 171,500 / 175,000 . Assuming your first application is a loser, there are still 3500 winners out of a remaining 174,999 tickets, so in that case your second application will be a loser with probability 171,499 / 174,999 . This continues down to your 100th application, which - if your first 99 are losers - will be a loser with probability 171,401 / 174,901 . Now you have to multiply all these numbers together. That shouldn't be too hard with a calculator, but a fast way to get an approximate answer is to notice that all these numbers are almost the same, so you could just take (171,500 / 175,000) to the power 100, = (.98)^100 . Now, if you know about the number e , you will recognize that (.98)^50 = (1 - 1/50)^50 is very close to 1 / e . footnote: e is approximately 2.718, and is the limit as N gets large of (1 - 1/N)^N . So (.98)^100 is the square of (.98)^50, so it is about 1 / e^2, which is approximately .135. Now remember, that's the chance of LOSING, so your chance of winning is really quite good: approximately 1 - .135 = .865. Hope this helps. Sorry the answer took so long, but you reached us at a time when we were shuffling our probability personnel (so to speak). Dr. Math |
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