Binomial Calculator: Online Statistical Table
Use the Binomial Calculator to
compute individual and cumulative binomial probabilities. For help in using the
calculator, read the FrequentlyAsked 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 frequentlyasked
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. Online help is just a mouse click away.
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.
Outcome, x 
Binomial probability, P(X = x) 
Cumulative probability, P(X < x) 
0 Heads 
0.125 
0.125 
1 Head 
0.375 
0.500 
2 Heads 
0.375 
0.875 
3 Heads 
0.125 
1.000 
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 r 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
1,000, the Binomial Calculator uses a normal distribution to estimate the
binomial probabilities. In most cases, this yields good results 
often accurate within 1 or 2 percent. The larger the number of
trials, the more accurately the normal distribution estimates binomial
probabilities.

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 successes 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 probabilities. For
example, the
probability of getting AT MOST 7 heads in 12 coin tosses is a cumulative
probability equal to 0.806.)

Suppose the probability that a college freshman will graduate is 0.6 Three college
freshmen are randomly selected. What is the probability that
at most two of these students will graduate?
Solution:
We know the following:

The number of trials is 3 (because we have 3 students).

The number of successes is 2.

The probability of success for any individual student 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 two or
fewer of these three students will graduate is 0.784.
(Note that the calculator also displays
the binomial probability  the probability that EXACTLY two of these students
graduate. The binomial probability is 0.432.)