Binomial Distribution

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A trial, which has only two outcomes i.e., "a success" or "a failure", is called a Bernoulli trial.

Let X be the number of successes in a Bernoulli trial, then X can take 0 or 1 and

P(X =1) = p = "probability of a success"

P(X = 0) = 1 - p = q = "probability of failure".

Consider a random experiment of performing n independent Bernoulli trials.

Let p be the probability of success, q = 1 - p be the probability of failure.

The probability of x successes and consequently (n - x) failures in n independent trials in a specified order say SSSFFSSFF….FSF is given by

P(SSSFFSSFF…FSF)

= P(S) P(S) P(S) P(F) P(F) P(S) P(S) P(F)….P(F) P(S) P(F)

= p.p.p.qq.ppq….qpq

= p^xq^n-x

But x successes can occur in ⁿC_x ways.

\ P(X = x) = ⁿC_x p^x q^n-x is the probability mass function of exactly x successes.

The probability distribution of the number of successes, so obtained is called the binomial distribution.

X P[X = x]

0 qⁿ

1 ⁿC₁ pq^n-1

2 ⁿC₂ p² q^n-2

3 ⁿC₃ p³ q^n-3

. .

. .

. .

. .

n pⁿ

Note 1:

n and p are called the parameters of the binomial distribution.

Note 2:

If x is a binomial variate with parameters n and p then it is denoted by x = b(n, p).

Note 3:

Example:

5 cards are drawn successively with replacement from well shuffled deck of 52 cards. What is the probability that

i) all the five cards are spades

ii) only 3 cards are spades

iii) none is a spade.

Suggested answer:

Let X be the random variable for the number of spade cards drawn.

p = probability of drawing a spade card

q = 1 - p

n = 5

Recurrence Relation for the Binomial Distribution

We have

P(X = x + 1) = ⁿC_x+1 p^x+1 q^n-x-1

Mean and variance

= np (p+q)^n-1

To find the variance:

We have

= n(n-1) p² (p+q)^n-2 + np

= n²p² - np² + np

= n²p² + np(1-p)

= n²p²+ npq

Now V(x) = E(x²) - [E(x)]²

= n²p² + npq - n² p²

= npq

Example:

If the mean and variance of a binomial distribution are respectively 9 and 6, find the distribution.

Suggested answer:

Mean of a binomial distribution = np = 9

Variance of a binomial distribution = npq = 6

Binomial distribution is given by