Binomial Distribution and Hypothesis Testing

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Currently being written. Last update on 29th January 2003.

The binomial distribution is a mathematical model for calculating the probability of a certain number of events, n, occuring when the probability of each event is p.

The Binomial Distribution

The model uses the equation: P(x=r) = nCrprqn-r where the probability that the event, x, occurs r times is being found; p is the probability of x occuring in each trial; q is the probability of x not occuring in each trial1 and n is the number of trials.

For example, if five ordinary dice are rolled and the probability of having four dice showing six is being found:

  • x represents the number of sixes rolled
  • r = 4 as the probability of four sixes being rolled is to be calculated
  • n = 5 as each die can be counted as being an independant trial -- one die's score does not affect the other dice
  • p = 1/6 - the dice are not weighted so the probability of any of the six numbers being rolled is 1/6
  • q = 5/6 as q = 1-p = 1-1/6 = 5/6

The following calculation can be performed:

    P(x=4) = 5C4 x (1/6)4 x (5/6)5-4

    P(x=4) = 5 x (1/1296) x (5/6)

    P(x=4) = 25/7776

    P(x=4) = 0.003215 (4sf)2

The probability of getting four "sixes" when five dice are rolled is 0.003215, or 0.3215%

Conditions for modelling using the binomial distribution

The binomial distribution model can only be used to approximate the probability of an event occuring a certain number of times during a set number of trials when the probability of the event occuring in each trial is known.

Hypothesis Testing

One of the most useful aspects of the binomial distribution is the ability to test, at a certain significance level, if the result obtained happened by chance or if it was likely not to have been chance. The theory behind this is that, at for example the 5% significance level, if the calculated probability of the obtained result is less than 5% then the result is too unlikely to have happened by chance. Two hypotheses can, therefore, be proposed:

  1. H0 - the null hypothesis, that the result happened by chance.
  2. H1 - the alternative hypothesis, that the result did not happen by chance.
1ie, q = 1-p24 significant figures

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