## Book: RD Sharma - Mathematics (Volume 2)

### Chapter: 33. Binomial Distribution

#### Subject: Maths - Class 12th

##### Q. No. 42 of Exercise 33.1

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42
##### Suppose X has a binomial distribution with n=6 and p=1/2. Show that X = 3 is the most likely outcome.

Given that, X has a binomial distribution.

Also, n = 6 & p = 1/2

And as we know that, p + q = 1

q = 1 – p

Let us define a binomial distribution with x as a variable out of other n variables.

P (X = x) = nCxpxqn-x

Putting values of n, p and q, we get

From the derived result, it can be seen that P (X = x) will be maximum if 6Cx is maximum.

Let us check at what value of x would 6Cx be maximum.

Check at x = 0.

6C0 = 1

Check at x = 6.

6C6 = 1

Check at x = 1.

6C1 = 6

Check at x = 5.

6C5 = 6

Check at x = 2.

6C2 = 15

Check at x = 4.

6C4 = 15

Check at x = 3.

6C3 = 20

Note that at x = 3, 6Cx is maximum.

Hence, we have shown that x = 3 is the most likely outcome.

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