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### Chances of Winning a Raffle

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Date: 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
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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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Associated Topics:
High School Probability

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