Probability Sequencing Riddle
Posted by Rob Herman at July 24th, 2007
Consider a (fair) six-sided die with four sides red, two sides blue. That is, it has a 2/3 chance to roll red, 1/3 blue. We are playing a game as follows: One of us picks a sequence of two outcomes (like “blue, red”), and then the other picks a different sequence of two outcomes. We roll the die as many times as necessary until the most recent two rolls match one of our sequences, at which point that player wins.
(Example: you pick red, red and I pick blue, blue. The rolls come up 1. red, 2. blue, 3. red, 4. blue, 5. red, 6. red, at which point you win.)
Because I am magnanimous, I will allow you to pick the sequence first or second, as you like. What is your winning strategy?
Edit: I should credit the source, which is here. I recast the riddle because I feel that the die is more intuitive than a “biased coin flip” and to try to avoid what I feel is a major hint in the wording of the original.
Unfortunately, I have to admit that I largely butchered elegance in the face of the relative simplicity of brute force calculation.
On the surface, one might expect to pick first and choose the most probable overall pairing (RR = 4/9). However, what intuition I did have immediately into the answer was that the power of tailoring my choice to your choice might do better.
First and easy point - if you choose RR, I choose BR, and I win all 5/9 games that start with any pair other than RR (once a blue has rolled, I win on the next red).
On to the option-crunching.
Symmetrically against a BB choice I take RB. 8/9 win chance.
Basically the idea is to have you rolling dead after the first two rolls, which I cannot stop you from winning on.
Against BR, I choose RB. You win when the first roll is B, I win when the first roll is R. 6/9 win chance.
Against RB, I choose RR. This one is slightly different to calculate. Since we both start with the same colour (only instance of this occurring), we can ignore all blue rolls until the first R. Having rolled a R, I win on R you win on B next roll. 6/9 win chance.
And thus I take the bread from your starving children’s mouths to line my pockets with gold.
I lied a little. Turns out once I got down to combatting each of your potential choices, my immediate intuition resulted in a winning strategy, so I didn’t write out the entire matrix of win%. Probably will just for the mental exercise later. Guess I haven’t lost as much of my edge as I generally assume.