Binomial Distribution Formula

About Binomial Distribution Formula

In statistics, the binomial distribution is a regularly used discrete distribution. In contrast to a binomial distribution, the normal distribution is a continuous distribution. Given a success probability of 'p' for each trial at the experiment, the binomial distribution represents the chance of 'x' successes in 'n' trials.

What Is the Binomial Distribution Formula?

For each random variable X, the binomial distribution formula is:

P(x :n,p) = nCxpx(1 - p)n-xor p(x : n,p) = nCxpx(q)n-x

Here,

  1. n = Number of experiments
  2. x = 0, 1, 2, 3, 4, …
  3. p = In a single experiment, Probability of success
  4. q = In a single experiment, Probability of failure (= 1 – p)

The n-Bernoulli trials are also used in the binomial distribution formula. Here

nCx = n!/ x!(n-x)!.Hence,p(x:np) = n!/[x!(n - x)!].px.(q)(n-x)

For x=2, P(x=2) = 5C2 p2 q5-2 = 5! / 2! 3! × (½)2× (½)3

P(x=2) = 5/16

For x ≥ 4,

p(x ≥ 4) = p(x = 4) + p(x = 5)

Thus, p(x = 4) = 5C4p4q5-4 = 5!/ 4! 1! × (½)4 ×(½)1 = 5 /32

p(x = 5) = 5C5p5q5-5=(½)5=1/32

Solved example of Binomial Distribution Formula

  1. Example: Find the probability of: if a coin is tossed 5 times using binomial distribution
    1. 2 heads exactly
    2. 4 or more heads.
  2. Sol: (a) A Bernoulli trial is exemplified by the repeated tossing of a coin. In light of the issue:
    1. Trials conducted: n=5
    2. Probability of head: p= 1/2
    3. Thus, the probability of tail, q =1/2
  3. For exactly two heads:
  4. (b) 4 or more heads,
  5. Answer: Therefore, P(x ≥ 4) = 5/32 + 1/32 = 6/32 = 3/16
  6. Example: For the same question given above, find the probability of getting at most 2 heads.
    1. P(at most 2 heads) = P(X ≤ 2) = P (X = 0) + P (X = 1)
    2. P(X = 0) = (½)5 = 1/32
    3. P(X=1) = 5C1 (½)5.= 5/32
    4. Answer: Therefore, P(X ≤ 2) = 1/32 + 5/32 = 3/16

To get all the Maths formulas check out the main page. 

Find pdf of Binomial Distribution Formula with solved examples 

Binomial Distribution Formula

Related Links

Frequently Asked Questions on Binomial Distribution Formula