Get inspired by the success stories of our students in IIT JAM MS, ISI  MStat, CMI MSc DS.  Learn More

# ISI MStat PSB 2006 Problem 5 | Binomial Distribution

This is a very beautiful sample problem from ISI MStat PSB 2006 Problem 5 based on use of binomial distribution . Let's give it a try !!

## Problem- ISI MStat PSB 2006 Problem 5

Suppose $$X$$ is the number of heads in 10 tossses of a fair coin. Given $$X=5,$$ what is the probability that the first head occured in the third toss?

### Prerequisites

Basic Counting Principle

Conditional Probability

Binomial Distribution

## Solution :

As $$X$$ is the number of heads in 10 tossses of a fair coin so $$X \sim binom(10, \frac{1}{2} )$$

A be the event that first head occured in third toss

B be the event that X=5

We have to find that $$P(A|B)=\frac{P(A \cap B)}{P(B)} = \frac{ {7 \choose 4} {\frac{1}{2}}^{10} }{ {10 \choose 5} {\frac{1}{2}}^{10}}$$

As , $$P(A \cap B)$$ = Probability that out of 5 heads occur at 10 tosses 1st head occur at 3rd throw

=Probability that first two tails $$\times$$ probability that 3rd one is head $$\times$$ probability that out of 7 toss 4 toss will give head

= $${\frac{1}{2}}^2 \times \frac{1}{2} \times {7 \choose 4} {\frac{1}{2}}^{7}$$

Hence our required probability is $$\frac{5}{36}$$

## Food For Thought

Under the same condition find the probability that X= 3 given 1st head obtained from 2nd throw .