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) = ⁿC_xp^xq^n-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 ⁶C_x is maximum.

Let us check at what value of x would ⁶C_x 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, ⁶C_x is maximum.

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