Drawing Prizes

Date: 07/06/2003 at 01:20:54
From: Bruce Mutch
Subject: Logic - probability

Recently my son said that he should have waited to draw a number at a 
drawing instead of being one of the first ones to draw, (all the 
prizes were numbered and put in a hat and people could draw whenever 
they wanted to). There were many lesser prizes and a few good ones. 
He thought that the probability of getting a good prize would go up 
as more prizes were drawn - since there were so many cheaper prizes 
they would be drawn first. 

At first I thought over the logic in my head and agreed, but then I 
went to paper and my results seemed to indicate that the probability 
stayed the same of getting a good prize no matter when you drew. I 
simplified the logic and reduced it to 4 cheap prizes and 1 good 
prize, and it seems that the probability stays at 1/5 no matter when 
you draw. I wrote a Visual Basic program to test this and have tested 
it with 100,000 drawings and they don't look too even. What do you 
think, does the probability of getting the good prize stay the same no 
matter when you draw?

Date: 07/08/2003 at 18:28:40
From: Doctor Achilles
Subject: Re: Logic - probability

Hi Bruce,

Thanks for writing to Dr. Math.

Your reasoning is exactly right. I'm not sure what's going on with 
the visual basic program, but it's possible that the random number 
generator that comes with it isn't very good.

Here's another way to think about it:

Let's do a somewhat simplified game where there are 4 cheap prizes, 1 
good prize, and 20 raffle tickets and 20 contestants. One contestant 
will get the good prize, 4 will get cheap prizes, and 15 will get 
nothing.

Let's say that everyone draws a raffle ticket that has the prize 
written on it. That is, there is one raffle ticket that says "good 
prize," one that says "cheap prize #1," one that says "cheap prize 
#2," one that says "cheap prize #3," one that says "cheap prize #4," 
and 15 that say "no prize."

But there's a twist. All the tickets are sealed in dark envelopes and 
none of the contestants can look in the envelope until everyone has 
drawn (if you look in your envelope, you forfeit your prize).

Does it matter what order you draw? Absolutely not. All we have done 
is randomly distribute the raffle tickets and we all learn our prizes 
at the same time.

Now, probability is tricky, and when you learn things, that changes 
your odds. So how are things different when you get to learn as each 
person draws whether they won or not? If you are watching someone 
draw and they don't get the good prize, then your chances of getting 
it (assuming you haven't already drawn) go up, but just *slightly*, 
but if they win, then you have *no* chance of winning.

For example, if you are watching the first person in a group of 20 
draw, and s/he doesn't get the good prize, then your odds just went 
up from 1/20 to 1/19. HOWEVER, if the first person does get the good 
prize then your odds go all the way from 1/20 down to 0. There is a 
1/20 chance that the first person will win and a 19/20 chance s/he 
will lose. So there is a 19/20 chance you will have a 1/19 chance. If 
you multiply those probabilities, you end up with a 1/20 chance. Add 
that to your 1/20 chance of 0 (which works out to 0) and you end up 
with a net probability of 1/20 after the first person draws: exactly 
the same as before.

I hope this helps. If you have other questions or you'd like to talk 
about this some more, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/