Probability of Never Having a Losing Record

Date: 04/07/2001 at 20:15:40
From: Omer Bulut
Subject: Probability

Suppose a football team plays 8 games. The chance of winning any 
particular game is 50%. Determine the probability of completing the 
season without ever having more losses than wins during the season.

It is really hard for me to do this question.

Thank you,
Omer

Date: 04/08/2001 at 12:38:53
From: Doctor Anthony
Subject: Re: Probability

This is by no means a trivial problem. You can model it as the number 
of paths on a grid of lattice points from the origin to (m,n) such 
that you never cross but may touch the line y = x. In the case when 
m = n we have what are known as Catalan numbers. If you are interested 
in seeing the proof of the formulae used below you should write again.

In the work that follows we are referring to ACCEPTABLE paths, that 
is, paths in which at all stages y > x, though we can accept y = x.

The required number of acceptable paths is given by:

     (n-m+1)
     -------.C(m+n,m)
       n+1

Using this model with 8 football matches, we have m+n = 8 and there is 
equal probability at each stage (or game) that either m or n will 
increase by 1.

Number of routes from the origin to (0,8) = (9/9).C(9,0) =  1

  "         "               "       (1,7) = (7/8).C(8,1) =  7

  "         "               "       (2,6) = (5/7).C(8,2) = 20

  "         "               "       (3,5) = (3/6).C(8,3) = 28

  "         "               "       (4,4) = (1/5).C(8,4) = 14
                                                  ------------
                                                   Total = 70

The total number of possible sequences of Win/Lose is 2^8 = 256 and so 
the required probability = 70/256
                         = 35/128   (=0.27344)
  
- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/