4. BERNOULLI TRIALS AND THE BINOMIAL DISTRIBUTION
So far in our discussion of probability we have learned about
combinations and permutations because they help us find the number
of ways a certain event can happen. Using that information we
calculate probabilities. Today we are learning a formula that is used
for very specific situations. We will start with a definition:
A Bernoulli experiment is a random experiment, the outcome of
which can be classified as either a success or failure
(e.g., female or male, life or death, non­defective or defective, heads
or tails, pass or fail).
A sequence of Bernoulli trials occurs when a Bernoulli experiment is
performed several independent times so that the probability of
success, p, remains the same from trial to trial.
If the probability of a success = p, and the probability of a failure = q
then q = 1­p because the probability of a success and failure must add
up to 1.
Binomial Distribution
In a sequence of Bernoulli trials we are often interested in the total
number of successes and not in the order of their occurrence. If we let
the random variable X equal the number of observed successes in n
Bernoulli trials, the possible values of X are 0,1,2,…,n. If x success
occur, where x=0,1,2,...,n , then n­x failures occur. The number of
ways of selecting x positions for the x successes in the x trials is: nCx
Citation:
http://cnx.org/content/m13123/latest/

5. These probabilities are called binomial probabilities, and the random
variable X is said to have a binomial distribution.
Summarizing,
a binomial distribution satisfies the following properties:
1. A Bernoulli (success­failure) experiment is performed n times.
2. The trials are independent.
3. The probability of success on each trial is a constant p; the
probability of failure is q =1−p .
4. The random variable X counts the number of successes in the n
trials.

6. The formula to find the probability in a binomial
distribution is: