First slide

Random variable and probability distribution

Question

A coin is tossed 2n times. The chance that the number of times one gets head is not equal to the number of times one gets tails is

Moderate

A

$\frac{(2 n!)}{(n!)^{2}} {(\frac{1}{2})}^{2 n}$
B

$1 - \frac{(2 n!)}{(n!)^{2}}$
C

$1 - \frac{(2 n!)}{(n!)^{2}} \frac{1}{4^{n}}$
D

none of these

Solution

The required probability is
1 - Probability of getting equal number of heads and tails
$\begin{array}{l} = 1 -^{2 n} C_{n} {(\frac{1}{2})}^{n} {(\frac{1}{2})}^{2 n - n} \\ = 1 - \frac{(2 n)!}{n! n!} {(\frac{1}{4})}^{n} \\ = 1 - \frac{(2 n)!}{(n!)^{2}} \times \frac{1}{4^{n}} \end{array}$

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