Question

In a gambling between Mr. $A$ and Mr. $B$, a machine continues tossing a fair coin until the two consecutive throws either $HT$ or $TT$ are obtained for the first time. If it is $HT$, Mr. $A$ wins and if it is $TT$, Mr. $B$ wins. Which of the following is(are) true?

• A. probability of winning Mr. $A$ is $\frac{3}{4}$
• B. probability of winning Mr. $B$ is $\frac{1}{4}$
• C. Given that first toss is head,probability of winning Mr. $A$ is $1$
• D. Given that first toss is tail,probability of winning Mr. $A$ is $\frac{1}{2}$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A probability of winning Mr. $A$ is $\frac{3}{4}$
B probability of winning Mr. $B$ is $\frac{1}{4}$
C Given that first toss is head,probability of winning Mr. $A$ is $1$
D Given that first toss is tail,probability of winning Mr. $A$ is $\frac{1}{2}$

If $T$ comes in first toss then Mr. $B$ can win in only one case that is $TT$.
Probability of mr. $B$ wining $=\frac{1}{4}$
Probability of mr. $A$ wining $=\frac{3}{4}$
Given first toss is head, Mr. $A$ can win in successive tosses are $T,HT,HHT........$
Probability $=\frac{\frac{1}{2}}{1-\frac{1}{2}}=1$
Given first toss is tail, Mr. $A$ can win in successive tosses are $HT,HHT,HHHT........$
Probability $=\frac{\frac{1}{4}}{1-\frac{1}{2}}=\frac{1}{2}$