Imagine that Alice and Bob play for a while. Alice can win after two rolls (if they are equal) but Bob can’t. Bob will complain that he doesn’t like it when the game ends early and Alice wins. Alice then makes the following generous offer: “I won’t claim a win if the first two rolls happen to be equal. When that happens, we’ll keep going until two later consecutive rolls match”. Alice and Bob are now playing a slightly different game which is certainly better for Bob than the old one.

Their friend Eve who is watching points out that since nobody will use the first dice roll to call an end to the game, there’s no need to know what it is until the game ends. After the first dice is rolled, Eve covers it with a cloth and reveals it after the game has ended and the other rolls are summed. When Eve uncovers the first dice and adds its score to whatever other total there is, it’s equally likely that it makes the overall total even or odd. Therefore in the new game Alice and Bob have equal chances to win. The old game is better for Alice and she is more likely to win.

]]>How I see it: we know that the last two rolls are the same, so the sum of the last two rolls is even. So the last two rolls don’t affect the parity of the total sum. Then, for any sequence of rolls the probability of sum being even or off is 1/2. The only tricky case not covered by the logic just described is when the first two rolls are the same, so only 2 rolls are made in total. In this case the sum is even. Thus, the chance of rolling even sum is higher due to this corner case.

]]>If A and B throw the same number the sum would be even which would keep the parity. So it seems they have fair chances but the first

roll is something else. If they roll the same number in the first roll A would win because the sum before would be the even 0.

So A’s chances to win are: P(2 equal numbers in the first roll) + 1/2 * P(no equal numbers in the first roll) = 1/6 + (1/2) * (5/6) = 7/12 ]]>

EE (the last outcome is even and the sum is even)

OO (the last outcome is odd and the sum is odd)

OE (the last outcome is odd and the sum is even)

EO (the last outcome is even and the sum is odd)

From EE and OO the game can be terminated in E (end of game with even total) while from OE and EO is O (end of game with odd total) that can be reached.

The first roll of the die places us either in EE or in OO (the sum equals the last outcome): it’s apparent that E is favoured.

PS: the odds for E are 47:35.

]]>