Dice game keep rolling

Question

There is a dice game where both players roll a fair dice once. If player A rolls a 1, then he keeps rolling. And in the end, if the score is tied, player B wins. What is the probability of player A winning?

Answer 1

Make yourself a grid of all possibilities:

$\begin{array}{|cccccc}\hline \color{blue}{2>1}&2\leq 2&2\leq 3&2\leq 4&2\leq 5&2\leq 6\\ \color{blue}{3>1}&\color{blue}{3>2}&3\leq 3&3\leq 4&3\leq 5&3\leq 6\\ \vdots\\ \color{blue}{6>1}&\dots&&\dots&\color{blue}{6>5}&6\leq 6\end{array}$

We count how many "good" possibilities there are. In the first row there is one, in the second row there are two, etc... on up until the last row which has five good possibilities giving a total of $1+2+3+4+5=\frac{5\cdot 6}{2}=15$ scenarios in which $A$ wins.

Notice that each of the outcomes in the grid are equally likely to occur. (This might be somewhat challenging to a beginner, but the hand-wavy explanation is that since $A$ is rerolling whenever he gets a $1$, he might as well have rolled a "five-sided die" instead which only has numbers $2$ through $6$.)

Now, taking the ratio of the number of "good" scenarios to the total number of scenarios gives the probability:

$$Pr(A~\text{wins})=\frac{15}{30}=\frac{1}{2}$$