Why is there no advantage of rolling a die first in chance games?

Question

Let's say if you roll 6, you win. The first player has a chance of winning 1/6. And then, the second player has a chance of winning (5/6)*(1/6) where 5/6 comes from the first player not rolling 6 in his turn, or so I thought. But they say that the second player has an equal chance of winning, 1/6. Why is it?

Answer 1

In fairness to you, assuming that the picture with the paragraph describing the rules is a screenshot from the problem as it was posed to you, that description of the rules is appallingly bad.

"You will lose the round if you or your opponent roll a 1-1, 2-2, 4-4 or a 2-1." No, according to everyone else, if your opponent rolls 1-1, 2-2, 4-4, or 2-1, your opponent loses and you win.

(Actually, none of the online sources I checked had a simple, ironclad description of the rules; everyone used some ambiguous language. But the description in the problem statement still stands out as the worst I found. The online Encyclopedia Britannica was the best.)

Assuming that there is no way to distinguish 6-5 from 5-6, when you roll there is a $\frac{5}{36}$ chance that you win immediately because you roll one of the winning rolls, a $\frac{5}{36}$ chance that you lose immediately because you roll one of the losing rolls, and a $\frac{26}{36}$ chance that the game simply continues.

When your opponent rolls, there is a $\frac{5}{36}$ chance that you win immediately because your opponent rolls one of the losing rolls, a $\frac{5}{36}$ chance that you lose immediately because your opponent rolls one of the winning rolls, and a $\frac{26}{36}$ chance that the game simply continues.

So it really does not matter who rolls when: every time the dice are rolled, you have the same chance to win or to lose.

How you would know this from the atrociously written problem statement is another question. Did you pay tuition for this kind of treatment? If so, perhaps you deserve at least a partial refund.

Answer 2

You changed the winning rule to "$6$ wins". If the first $6$ wins then you should go first.

But if the rule is "$6$ wins and $1$ loses" then the game is like the one in the picture. You win or lose with the same probability ($1/6$) on each roll so it does not matter if you go first or second.