Puzzle. Alice and Bob roll an unbiased 6-sided die until two consecutive rolls are the same. They add up the scores from all of the rolls. If the total is even, Alice wins; if it is odd, Bob wins. Who is more likely to win?

Interested to know if there’s an “aha” solution to this ðŸ™‚ A variant, where they keep going until two consecutive rolls are *different* (i.e., until two different results have been observed) seems easier.

We can track the game by recording the last outcome and the sum. Both of them are either even or odd so we have four possibilities:

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.

## KK:

Alice

19 May 2023, 4:44 pm## James:

Interested to know if there’s an “aha” solution to this ðŸ™‚ A variant, where they keep going until two consecutive rolls are *different* (i.e., until two different results have been observed) seems easier.

21 May 2023, 7:12 pm## Guido:

We can track the game by recording the last outcome and the sum. Both of them are either even or odd so we have four possibilities:

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.

26 May 2023, 5:47 am