Coin vs Die

John makes a bet with Donald that he’s able to flip heads on a coin before Donald throws 2 in a die. What’s the probability of Donald winning, considering John begins the game and they play alternately?

Let the probability of tossing an head be $p=~^1/_2$ and tail $q=~^1/_2$ , and the probability of tossing any die face other than a 2 be $r=1-~^1/_6=~^5/_6$ . If we consider the outcome of John winning, then:

outcome	probability
$H$	$p$
$T⚅H$	$qrp$
$T⚅T⚅H$	$qrqrp$
…	$(qr)^np$

Therefore, the converse probability is given by:

$1 - \sum_{n~=~0}^{\infty}{^1/_2~\left(^5/_6\cdot~^1/_2\right)^n}$

Since this is a geometric series with $\lvert~r~\rvert =~^5/_6\cdot~^1/_2 < 1$ , then $\sum_{n~=~0}^{\infty}a_i=\frac{a}{1-r}$ , so:

$\begin{align} 1 - \sum_{n~=~0}^{\infty}{^1/_2~\left(^5/_6\cdot~^1/_2\right)^n}&=1-\frac{^1/_ 2}{1-~^5/_{12}}\\ &=~^1/_7 \end{align}$