Binomial Distribution Calculator

Use the Binomial Calculator to compute individual and cumulative binomial probabilities. For help in using the calculator, read the Frequently-Asked Questions or review the Sample Problems.

To learn more about the binomial distribution, go to Stat Trek's tutorial on the binomial distribution.

Instructions: To find the answer to a frequently-asked question, simply click on the question. If none of the questions addresses your need, refer to Stat Trek's tutorial on the binomial distribution or visit the Statistics Glossary. On-line help is just a mouse click away.

• What is a binomial experiment?
• What is a binomial distribution?
• What is the number of trials?
• What is the number of successes?
• What is the probability of success on a single trial?
• What is the binomial probability?
• What is the cumulative binomial probability?
• What is the relation between the binomial and normal distributions?

What is a binomial experiment?

A binomial experiment has the following characteristics:

• The experiment involves repeated trials.
• Each trial has only two possible outcomes - a success or a failure.
• The probability that a particular outcome will occur on any given trial is constant.
• All of the trials in the experiment are independent.

A series of coin tosses is a perfect example of a binomial experiment. Suppose we toss a coin three times. Each coin flip represents a trial, so this experiment would have 3 trials. Each coin flip also has only two possible outcomes - a Head or a Tail. We could call a Head a success; and a Tail, a failure. The probability of a success on any given coin flip would be constant (i.e., 50%). And finally, the outcome on any coin flip is not affected by previous or succeeding coin flips; so the trials in the experiment are independent.

What is a binomial distribution?

A binomial distribution is a probability distribution. It refers to the probabilities associated with the number of successes in a binomial experiment.

For example, suppose we toss a coin three times and suppose we define Heads as a success. This binomial experiment has four possible outcomes: 0 Heads, 1 Head, 2 Heads, or 3 Heads. The probabilities associated with each possible outcome are an example of a binomial distribution, as shown below.

What is the number of trials?

The number of trials refers to the number of attempts in a binomial experiment. The number of trials is equal to the number of successes plus the number of failures.

Suppose that we conduct the following binomial experiment. We flip a coin and count the number of Heads. In this experiment, Heads would be classified as success; tails, as failure. If we flip the coin 3 times, then 3 is the number of trials. If we flip it 20 times, then 20 is the number of trials.

What is the number of successes?

Each trial in a binomial experiment can have one of two outcomes. The experimenter classifies one outcome as a success; and the other, as a failure. The number of successes in a binomial experient is the number of trials that result in an outcome classified as a success.

What is the probability of success on a single trial?

In a binomial experiment, the probability of success on any individual trial is constant. For example, the probability of getting Heads on a single coin flip is always 0.50. If "getting Heads" is defined as success, the probability of success on a single trial would be 0.50.

What is the binomial probability?

A binomial probability refers to the probability of getting EXACTLY n successes in a specific number of trials. For instance, we might ask: What is the probability of getting EXACTLY 2 Heads in 3 coin tosses. That probability (0.375) would be an example of a binomial probability.

What is the cumulative binomial probability?

Cumulative binomial probability refers to the probability that the value of a binomial random variable falls within a specified range.

The probability of getting AT MOST 2 Heads in 3 coin tosses is an example of a cumulative probability. It is equal to the probability of getting 0 heads (0.125) plus the probability of getting 1 head (0.375) plus the probability of getting 2 heads (0.375). Thus, the cumulative probability of getting AT MOST 2 Heads in 3 coin tosses is equal to 0.875.

Notation associated with cumulative binomial probability is best explained through illustration. The probability of getting FEWER THAN 2 successes is indicated by P(X < 2); the probability of getting AT MOST 2 successes is indicated by P(X < 2); the probability of getting AT LEAST 2 successes is indicated by P(X > 2); the probability of getting MORE THAN 2 successes is indicated by P(X > 2).

What is the relation between the binomial and normal distributions?

When the number of trials is large and when the probability of success is not extreme (i.e., neither close to 0 nor close to 1), then the normal distribution may be used to very closely approximate results from the binomial distribution.

Note: When the number of trials is greater than 20,000, the Binomial Calculator uses a normal distribution to estimate the cumulative binomial probability. In most cases, this yields very good results - often accurate to the third decimal place.

1. Suppose you toss a fair coin 12 times. What is the probability of getting exactly 7 Heads.
   
   Solution:
   
   We know the following:
   
   
   • The number of trials is 12.
   • The number of success is 7 (since we define getting a Head as success).
   • The probability of success (i.e., getting a Head) on any single trial is 0.5.
   
   Therefore, we plug those numbers into the Binomial Calculator and hit the Calculate button. The calculator reports that the binomial probability is 0.193. That is the probability of getting EXACTLY 7 Heads in 12 coin tosses. (The calculator also reports the cumulative probability - the probability of getting AT MOST 7 heads in 12 coin tosses. The cumulative probability is 0.806.)
   
   
2. Suppose the probability that a college freshman will graduate is 0.6 Three sisters (triplets) enter college at the same time. What is the probability that at most 2 sisters will graduate?
   
   Solution:
   
   We know the following:
   
   
   • The number of trials is 3 (because we have 3 sisters).
   • The number of successes is 2.
   • The probability of success for any individual sister is 0.6.
   
   Therefore, we plug those numbers into the Binomial Calculator and hit the Calculate button. The calculator reports that the cumulative binomial probability is 0.784. That is the probability that 2 or fewer sisters will graduate is 0.784. (Note that the calculator also displays the binomial probability - the probability that EXACTLY 2 sisters graduate. The binomial probability is 0.432.)