The Binomial

Binomial experiments

Binomial experiments require the following elements:

• The experiment consists of a number of identical events ( n).

• Each event has only one of two mutually exclusive outcomes. (These outcomes are called successes and failures.)

• The probability of a success outcome is equal to some percentage, which is identified as a proportion, π.

• This proportion, π, remains constant throughout all events and is defined as the ratio of number of successes to number of trials.

• The events are independent.

• Given all of the above, the binomial formula can be applied ( x = number of favorable outcomes; n = number of events):
  

  
  

Example 1: A coin is flipped 10 times. What is the probability of getting exactly 5 heads? Using the binomial formula, where n (the number of events) is given as 10; x (the number of favorable outcomes) is given as 5; and the probability of landing a head in one flip is .5:

So the probability of getting exactly 5 heads in 10 flips is .246, or approximately 25 percent.

Binomial table

Because probabilities of binomial variables are so common in statistics, tables are used to alleviate having to continually use the formula.

Mean and standard deviation

The mean of the binomial probability distribution is determined by the following formula:

The standard deviation of the binomial probability distribution is determined by this formula:

Example 2: What is the mean and standard deviation for a binomial probability distribution for 10 coin flips of a fair coin?

Because the proportion of favorable outcomes of a fair coin falling heads (or tails) is π = .5, simply substitute into the formulas:

Note that this distribution appears to display symmetry.