Question

A fair coin is tossed n times. The probability that the difference between the number of heads and tails in (n - 3) is

• A. $2^{-n}$
• B. 0
• C. ${}^n{C}_{n-3}{2}^{-n}$
• D. $2^{-n+3}$

Answer 1

The correct option is B 0

Let no. of heads = x
no. of tails = y
Given x + y = n
|x - y| = n - 3
|x - (n - x)| = n - 3
|2x - n| = n - 3
$\Rightarrow (2x-n)=\pm (n-3)$

Case I:
$\Rightarrow 2x-n=n-3$
$2x=2n-3$
$x=\frac{2n-3}{2}$

(not possible, as x and n are natural numbers)

Case II:
$2x-n=-n+3$

$\Rightarrow x=\frac{3}{2}\left(\text{Not possible}\right)$

$\therefore \text{Required probability}=0$