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?

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Probability of winning Mr.A is $\frac{3}{4}$

Probability of Mr.B winning is $\frac{1}{4}$

Given first toss is head probability of Mr. A winnings is 1

Given first toss is tail, probability of Mr.A winning is $\frac{1}{2}$

answer is A, B, C, D.

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Detailed Solution

If T comes in first toss then Mr. B can win in only one case that is TT.
$\Rightarrow$ probability of Mr. B winning = $\frac{1}{4}$

$\Rightarrow$ Probability of Mr.A winning = $\frac{3}{4}$
Given first toss is head, Mr. A can win is successive tosses are T,HT, HHT, …..
Probability $= \frac{\frac{1}{2}}{1 - \frac{1}{2}} = 1$
Given first toss is head, Mr.A can win is successive tosses are HT, HHT, HHHT,…..
Probability $= \frac{\frac{1}{4}}{1 - \frac{1}{2}} = \frac{1}{2}$