Question

A fair coin is tossed $99$ times. If X is the number of times heads occur then P(X = r) is maximum when r is

• A. $49$
• B. $50$
• C. $51$
• D. none of these

Answer 1

Answer: (A)

$49$

Notice that x has binomial distribution with n = 99, p = q = 0.5.

Therefore the probability

$P(x=r){=}^n{C}_r{P}^r{q}^{n-r}⟹P(x=r){=}^n{C}_r\left(0.5{)}^r\right(0.5{)}^{n-r}{=}^n{C}_r(0.5{)}^n$

Thus maximum of P(x = r) corresponds to the value of r with maximal

${}^n{C}_r=\frac{n!}{r!(n-r)!}$

Notice that binomial coefficients are symmetric:

${}^n{C}_r{=}^{n-r}{C}_r=\frac{n!}{r!(n-r)!}$

hence the maximum corresponds to the points most closest to $\frac{n}{2}$

Since n is odd, there are two such values:

$r_1=\frac{n-1}{2}=\frac{98}{2}=49,r_2=\frac{n+1}{2}=\frac{100}{2}=50$

For these numbers the function P(x = r) has maximum.