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Binomial Probability Formula

The binomial distribution is a type of distribution in statistics that has two possible outcomes. For instance, if you toss a coin and there are only two possible outcomes: heads or tails. In the same way, taking a test could have two possible outcomes: pass or fail.

The binomial probability formula can calculate the probability of success for binomial distributions. The Binomial Probability distribution is an experiment that posses the following properties:

  • There are fixed number of trials which is denoted by n
  • All the trials are independent.
  • The outcome of each trial can either be a “success” or “failure”.
  • The probability of success remains constant and is denoted by p.

The Binomial Probability distribution of exactly x successes from n number of trials is given by the below formula-

\[\LARGE P(X) = C_{x}^{n} P^{x} q^{n-x}\]

Where,
n = Total number of trials
x = Total number of successful trials
p = probability of success in a single trial
q = probability of failure in a single trial = 1-p
Binomial Probability : Few solved examples
Example 1: A coin is flipped 6 times. What is the probability of getting exactly 2 tails.
Solution:
n = total number of trials = 6
x = total number of successful trials = 3
p = probability of success in one trial = $\frac{1}{2}$
q = probability of failure in one trial = 1 – $\frac{1}{2}$ = $\frac{1}{2}$

P(X) = $C_{x}^{n}$ $P^{x}$ $q^{n-x}$
P(x) = $\frac{6!}{2!\times 4!}$ $\times$ ($\frac{1}{2})^{3}$ $\times$ ($\frac{1}{2})^{6-3}$
= $\frac{6!}{2!\times 4!}$ $\times$ ($\frac{1}{2})^{3}$ $\times$ ($\frac{1}{2})^{3}$
= $\frac{15}{64}$
= 0.234