Binomial Distributions:
 Binomial distribution is the discrete probability distribution.
 Binomial distribution describes the possible number of times that a particular event will occur in a sequence of observations. The event is coded binary that is it may or may not occur.
 The binomial distribution is used when a researcher is interested in the occurrence of an event, not in its magnitude.
 All of the trials are independent and have only two possible outcomes, success or failure.
 The probability of success is the same in every trial.
 The outcome of one trial does not affect the probability of any future trials.
 The random variable is the number of success in a given number of trials.
 The probability of x successes in n independent trials is
P(x) = C(n,x)p^{x}q^{nx}
where
p is the probability of success of an individual trial
q is the probability of failure on that same individual trial (p + q = 1)
The expectation for a binomial distribution is
E(x) = np
where
n is the total number of trials and
p is the probability of success
Example:
If a coin is tossed 4 times, then we may obtain 0, 1, 2, 3, or 4 heads. We may also obtain 4, 3, 2, 1, or 0 tails, but these outcomes are equivalent to 0, 1, 2, 3, or 4 heads.
The likelihood of obtaining 0, 1, 2, 3, or 4 heads is, respectively, 1/16, 4/16, 6/16, 4/16, and 1/16.
p = 1/2 Thus, in the example discussed here, one is likely to obtain 2 heads in 4 tosses, since this outcome has the highest probability.
Directions: Answer the following questions. Also write at least 5 examples of Binomial distributions of your own.
