Question

Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are the same. If the probability of a random toss resulting in a head is $\frac{1}{3}$, then the probability that the experiment stops with the head is

• A. $\frac{1}{3}$
• B. $\frac{5}{21}$
• C. $\frac{4}{21}$
• D. $\frac{2}{7}$

Answer 1

Answer: (B)

$\frac{5}{21}$

Answer 2

The correct option is: (B) $\frac{5}{21}$
P(H) = $\frac{1}{3}$
P(T) = $\frac{2}{3}$
P(E) = P(HH) + P(THH) + P(HTHH) + P(THTHH) + P(HTHTHH) + P(THTHTHH) + .....
= $\frac{1}{3^2}+\frac{2}{3^3}+\frac{2}{3^4}+\frac{4}{3^5}+\frac{4}{3^6}+\frac{8}{3^7}+\frac{8}{3^8}+.......$
$=(\frac{1}{3^2}+\frac{2}{3^4}+\frac{4}{3^6}+.......)+(\frac{2}{3^3}+\frac{4}{3^5}+\frac{8}{3^7}+.......)$
P(E) = $\frac{1}{7}+\frac{2}{21}=\frac{5}{21}$