The Binomial

A discrete variable that can result in only one of two outcomes is called binomial. For example, a coin flip is a binomial variable, but drawing a card from a standard deck of 52 is not. Whether a drug is successful or unsuccessful in producing results is a binomial variable, as is whether a machine produces perfect or imperfect widgets.

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


Example 1

A coin is flipped ten times. What is the probability of getting exactly five 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 0.5: 

So, the probability of getting exactly five heads in ten flips is 0.246, 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. Refer to Table 1 in "Statistics Tables," and you will find that given n = 10, x = 5, and π = 0.5, the probability is 0.2461.

Mean and standard deviation

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

μ = nπ

where π is the proportion of favorable outcomes and n is the number of events.

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

What is the mean and standard deviation for a binomial probability distribution for ten coin flips of a fair coin?

Because the proportion of favorable outcomes of a fair coin falling heads (or tails) is π = 0.5, simply substitute into the formulas:


The probability distribution for the number of favorable outcomes is shown in Figure 1.

Note that this distribution appears to display symmetry. Only a binomial distribution with π = 0.5 will be truly symmetric. All other binomial distribution will be skewed.

Figure 1.The binomial probability distribution of the number of heads resulting from ten coin tosses.