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### Binomial Distribution and Probability

```
Date: 06/07/97 at 11:06:33
From: Bianca Clarke
Subject: Binomial distribution

I just don't understand binomial distribution. Can someone explain it
very simply?

Many thanks,
Bianca Clarke
```

```
Date: 06/07/97 at 15:53:26
From: Doctor Anthony
Subject: Re: Binomial distribution

Bianca,

The binomial distribution applies when you have repeated experiments
with constant probabilities of success and failure at each trial.  A
good example is throwing a die, say five times, and finding the
probability of rolling exactly three sixes.

If we use the letters S and F to represent success and failure in any
trial, then one possible sequence giving the desired result would be:

SSSFF

The probability of this particular sequence is:

(1/6)^3*(5/6)^2

Another sequence would be FFSSS and this too has probability:

(1/6)^3*(5/6)^2

In fact there are many other sequences that produce 3 sixes and 2 non-
sixes, and to get the total probability of 3 sixes and 2 non-sixes, we
must multiply the probability of any given sequence, (1/6)^3*(5/6)^2,
by the number of sequences.  This in effect, is the number of
different arrangements you can make with the 5 letters S and F, 3
being alike of one kind and two being alike of a second kind.

In case you are unfamiliar with this problem, we shall show that the
number of different arrangements is:

5!
------
3! 2!

We have 5 letters, and to start with assume they are all different
letters.

These letters could then be arranged in 5! ways. However three are
alike of one kind and we could swap these three amongst themselves in
3! ways without giving rise to a new arrangement.  Similarly two
others are alike of a second kind, and these could be swapped between
themselves without giving rise to a new arrangement.  Our answer of 5!
is therefore too large by factors 3! and 2!.  Hence the actual number
of different arrangements of three S's and two F's is given by:

5!
------   = 10
3! 2!

This is also the expression for the binomial coefficient 5_C_2.  These
coefficients you will find are given on most scientific calculators.

So the probability of three successes = 10*(1/6)^3*(5/6)^2

The general expression for binomial probabilities are given by the
terms of the binomial expansion of (p+q)^n

Here, n = number of trials
p = probability of success at each trial (constant)
q = probability of failure at each trial (p+q = 1)

The probability of r successes is P(r) = nCr*p^r*q^(n-r)

In example we did above, n = 5, p = 1/6, q = 5/6, and r = 3:

P(3) = 5C3*(1/6)^3*(5/6)^2
= 10*(1/6)^3*(5/6)^2  = 0.03215

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
High School Probability

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