A fair coin is tossed k times. If the probability of getting exactly six tails is equal to the probability of getting exactly 8 tails, then the probability of getting exactly 3 tails is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (2) → $\frac{37}{2^{12}}$
$P(T=6) = P(T=8)$
$\frac{k!}{6!(k-6)!} = \frac{k!}{8!(k-8)!}$
$\frac{1}{6!(k-6)!} = \frac{1}{8!(k-8)!}$
$8!(k-8)! = 6!(k-6)!$
$8 \cdot 7 \cdot 6!(k-8)! = 6!(k-6)(k-7)(k-8)!$
$56 = (k-6)(k-7)$
$k^2 -13k +42 = 56$
$k^2 -13k -14 = 0$
$k = 14$
$= \frac{14 \cdot 13 \cdot 12}{6 \cdot 2^{14}} = \frac{364}{16384}$
$= \frac{91}{4096}$
$\text{Required probability} = \frac{91}{4096}$