In probability theory, the Binomial distribution is a discrete probability distribution of a number of successes in a sequence of n independent experiments with a boolean value outcome: yes-no, head-tail, on-Off, True-False. In this p probability represents – success/yes/head/on/true **or** failure/tail/off/false represents probability (q = 1-p).

In simpler terms, binomial distribution ( bi means 2), has only two discrete mutually exclusive outcomes of an experiment.

E.g., tossing a coin has two possible outcomes – Tail or Head

**Probability Mass Function** – If the random variable X follows a binomial distribution with a total number of trails is equal to n and p [0,1]. Then X ~ B(n,p)

1. The probability of success in a single

2. The probability of getting exactly k successes in n trials is given by

3. Assuming p is fixed for all trails

**f(k,n,p) = Pr(X = k) = (n!)(pk)(1-p)n-k /(k!)(n-k)!**

**Let’s look at a couple of examples –**

1) Probability of getting exactly 5 heads in 20 coin flips

n – 20

k – 5

p = 0.5

f(5,20,.5) = Pr(X = 5) = 20!*(0.5)5(1-0.5)20-5/5!*(20-5)!

=** 0.147**

2) Probability of getting 2 exactly three times on rolling a die – 10 times

n – 10

k – 3

p = 1/6

f(3,10,1/6) = Pr(X = 3) =10!*(0.1/6)3(1-1/6)10-3/3!*(10-3)!

Using python – **Binomial Probability Mass Function**

**binom.pmf(3,10,1/6) = 0.155**

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