Binomial Probability Formula

The binomial probability formula can be used to calculate the probability of success for binomial distributions. Binomial probability distribution along with normal probability distribution are the two probability distribution types. To recall, 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 distribution is an experiment that possesses the following properties:

  • There are a 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.

Binomial Probability Formula

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

P (X) = nCx px qn – 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

Solved Examples For Binomial Probability

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 = 2

p = probability of success in one trial = 1/2

q = probability of failure in one trial = 1 – 1/2 = 1/2

P (X) = nCx Px qn – x

P (X) = [6!⁄(2! × 4!)] × (½)3 × (½)6 – 3

= [6!⁄(2! × 4!)] × (½)3 × (½)3

= 15⁄64

= 0.234

Example 2: Find the binomial distribution of random variable r = 4 if n = 10 and p = 0.4.

Solution:
Given,
n = 10
p = 0.4
q = 1 – p = 1 – 0.4 = 0.6
r = 4
Using the binomial probability distribution formula,
P(X = 4) = 10C4 p4 q10-4
= 10C4 (0.4)4(0.6)6
= (10!/4! 6!) × 0.0256 × 0.046656
= 210 × 0.0012
= 0.25 (approx)

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