Question

A fair coin is tossed $99$ times. Let X be the number of times heads occurs. Then $P(X=r)$ is maximum when r is

• A. $49$
• B. $52$
• C. $51$
• D. None of these

Answer 1

The correct option is A $49$

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

Therefore the probability,

$P(x=r){=}^r{C}_n{p}^r{q}^{n-r}$

Since $p=q=0.5$ we obtain that

$P(x=r){=}^r{C}_{99}\left(0.5{)}^r\right(0.5{)}^{99-r}{=}^r{C}_{99}(o.5{)}^{99}$

whence 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.