The binomial probability refers to the probability that an n-trial binomial experiment results in exactly 'x' successes, when the probability of success for each trial is 'p'. This probability is denoted by B(n, p, x). The formula for calculating the binomial probability is given as:
B(n, p, x) = P(X = x) = ,
where is equal to .
Cumulative binomial probability
A cumulative binomial probability refers to the probability that the binomial random variable falls within a specified range (e.g., is greater than or equal to a stated lower limit and less than or equal to a stated upper limit).
For example, we might be interested in the cumulative binomial probability of obtaining 4 or fewer heads in 10 tosses of a coin, that is P(X ≤ 4). This would be given as the sum of all the individual binomial probabilities below 4
i.e., P(X ≤ 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4).
Note: (i) If X is a binomial random variable, then X can take on the values 0, 1, 2, . . ., n.
(ii) If X is a geometric random variable, then it takes on the values 1, 2, 3, ...
(iii) There can be zero successes in a binomial, but the earliest a first success can come in a geometric setting is on the first trial.
Normal approximation to the binomial distribution: Many practical applications of the binomial distribution involve examples in which 'n' [i.e., sample size] is large. However, for large 'n', the binomial probabilities can be tedious to calculate. Since the normal distribution can be viewed as a limiting case of the binomial distribution, it is natural to use the normal to approximate the binomial in appropriate situations.
Binomial distribution is a discrete probability distribution. Hence it takes values only at integers whereas the normal distribution is a continuous probability distribution, hence it is continuous with probabilities corresponding to areas over interval. Therefore, we establish a technique for converting from one distribution to other distribution. (An approximation: Each binomial probability corresponds to the normal probability over a unit interval centered at the desired value.)
A normal distribution is a good approximation to the binomial distribution whenever both 'np' [or mean of the binomial distribution] and n(1 – p) are greater than or equal to 10
i.e., np ≥ 10 and n(1 – p) ≥ 10, where p = probability of success.