Suppose X has a binomial distribution . Show that X = 3 is the most likely outcome.

(Hint: P(X = 3) is the maximum among all P(xi), xi = 0,1,2,3,4,5,6)

As per the question,

X is any random variable whose binomial distribution is

Thus, P(X = x) = ⁿC_xq^n-xp^x , where x = 0, 1, 2, …n

It can be clearly observed that P(X = x) will be maximum if ⁶c_x will bw maximum.

∴⁶c_x = ⁶c₆ = 1

⁶c₁ = ⁶c₅ = 6

⁶c₂ = ⁶c₄ = 15

⁶c₃ = 20

Hence we can clearly see that ⁶c₃ is maximum.

∴for x = 3, P(X = x) is maximum.

Hence, proved that the most likely outcome is x = 3.