Section 2 : The Binomial Distribution(1)
 

[definition][notation][limits]

 

 

 

 

Definition

The Binomial Distribution describes the behavior of a random variable(count variable)  X under the following conditions:

1. the number of trials(n) is fixed
2. trials have only 2 possible outcomes (success/failure)
3. each trial is independent of the other
4. probability of success(p) is constant throughout

X(the random variable) is a measure of the number of successes in n trials.

Example

A simple example is choosing 1 ball from a bag of 10 identical balls, each numbered (1-10). Once noted, the ball is returned to the bag.
A single ball is chosen on 3 separate occasions.

Success is in obtaining a '5' ball.

So the random variable X has values 0, 1, 2, 3

in other words, from our 3 tries we could have obtained:

0 fives,    1 five,   2 fives,   3 fives  

On the first try, the probability of obtaining a 5 is 1/10 .
The probability of not getting a 5 is 9/10 .

Every time we dip into the bag of 10 balls, the probability of obtaining a '5' is 1/10. The probability is constant.

Getting a '5' or not getting a '5' means that there are only 2 outcomes.

Every try is independent. Previous tries do not affect the result, since previously chosen balls are returned to the bag. Every try is taken from a bag of 10 balls.

 

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Notation B(n, p)

The full notation describing a Binomial distribution is:

binomial distribution notation

where,

X a random variable (0, 1, 2, 3,...)
~ B   'is distributed Binomially'
n number of trials
p probability of single trial 'success'

Example (continued from above)

Say that there are only 3 tries of attempting to take a 5-ball from a bag of 10 balls.

So n = 3 and p = 1/10.

The possible number of 5's taken in the 3 trials is summarized by the values of the random variable X .

X = 0, 1, 2, 3

using the Binomial notation,

Binomial notation problem #1

 

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Limits

The population size(n) of a Binomial Distribution must be much larger than the sample size(r).

The distribution only applies to trials from a simple random sample, where n is at least x10 times > r .

Outside this limit, results do not follow the equation.

 

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