Re: [GZG] [OT] Probability formula needed

On Thu, Nov 17, 2005 at 08:45:07AM -0600, Allan Goodall wrote:

>I need a probability formula. The situation is as follows:
>
>You have 7 objects owned by 7 people. If you were to randomly guess
>which person owns which object, what is the probability that you would
>guess all 7 objects correctly? Guess 5 out of 7 correctly? Guess 4 out
>of 7 correctly?

The number of ways of choosing k objects from a set of n objects, where
the order matters, is n!/(n-k)! (conventionally written as nPk). So if
you want to get 7 right, there are 7!/(7-7)! or 5040 ways you can guess,
of which one is correct.

It gets a bit trickier for other numbers. The short-cut is to construct
Pascal's triangle:

    1
   1 1
  1 2 1
 1 3 3 1
1 4 6 4 1

etc. where each number is the sum of the two diagonally above it. The
relevant row is n=7:

1 7 21 35 35 21 7 1

So if you want to know the probability of picking 5 out of 7 correctly:

(1) if you wanted to pick the _first_ 5 out of 7, that would be 7P5 or
2525. So your chance of doing that is 1/2525.

(2) But it's not just the first five; different ways of doing that.
You might get (1 for right, 0 for wrong):

1 1 1 1 1 0 0
1 1 1 1 0 0 1
1 1 1 0 0 1 1
1 1 1 0 1 0 1
etc.

See the triangle: counting up from 0, column 5 has 21 in it, which tells
you that there are 21 different variations without you having to work
them all out. So your actual chance is higher: 21/2525 or about 1/120.

http://mathworld.wolfram.com/Permutation.html
http://mathworld.wolfram.com/PascalsTriangle.html

Roger
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