Associated Topics || Dr. Math Home || Search Dr. Math

### 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/
```
Associated Topics:
College Probability
High School Permutations and Combinations
High School Probability

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search