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$A$ and $B$ play a game of tossing a fair coin. $A$ starts the game by tossing the coin once and $B$ then tosses the coin twice followed by $A$ tossing the coin once and $B$ tossing the coin twice. This continues until a head turns up. Whoever gets the first head wins the game. Which of the following is/are true.

$(A)$ $P(B \ wins) > P(A \ wins)$

$(B)$ $P(B \ wins) = 2P(A \ wins)$

$(C)$ $P(B \ wins) < P(A \ wins)$

$(D)$ $P(A \ wins) = 1-P(B \ wins)$

I feel that $A,B,D$, are correct by intuition but I am not sure how to approach this. Any help will be greatly appreciated.

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    If there's a single correct answer then it is obviosly D. – 2017-01-07
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    @barakmanos No. There could be multiple correct options. – 2017-01-07
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    Just for intuition: $A$ can win on the very first toss, and might possibly win on a later toss. Thus it is clear that $P(A)>\frac 12$. – 2017-01-07

3 Answers 3

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$D$ is obviously correct.

Calculate the probability that $A$ wins, this happens under the following situations: head comes first try, then head comes on the fourth toss and tails on the rest(A gets tails on his toss, and $B$ on his tosses), then head on the seventh toss and tails on the rest, head on the tenth toss and tails on the rest etc.

Hence, the probability that $A$ wins is :$ \frac 12 + \left( \frac 12 \right)^4 + \left( \frac 12 \right)^7 + \ldots$ Compute this value using the geometric series formula, and use $D$ to find the probability that $B$ wins. This will help you see which option is correct and which is not.

In our case, it is even clearer : $P(A) > \frac 12$, since $A$ will win on the first turn with probability half, and on further turns with positive probability. Therefore, in light of $D$, one may rule out $A$ and $B$, and see that $C$ is obviously true.

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Hint:

These are the sequences that $A$ can win:

$H, TTTH, TTTTTTH, TTTTTTTTTH$

The probability is

$$\frac12+ \left(\frac12\right)^3 \frac12+\left(\frac12\right)^6 \frac12+\ldots> \frac12$$

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Without any work, it's instant that

  • (D) is true: With probability 1, a head eventually appears, so A wins or B wins, but (of course) not both.

  • (C) is true: Player A can win on the first toss, but can also possibly win after the first toss, so P(A) > 1/2.

  • (A) is false (since (C) is true).

  • (B) is false (since (A) is false).