easy Probability

Sum First I

A release-checked easy problem for training Conditional Expectation, Expected Value/LOTUS, Dice, Markov Chains.

Question

Alice and Bob alternate rolling pairs of standard fair 6-sided dice, with Alice going first. The game ends when Alice rolls a sum of $6$ or when Bob rolls a sum of $7$, whichever comes first. A turn is when either Alice or Bob rolls. Find the expected number of turns in the game.

Practice focus

This Probability problem is tagged Conditional Expectation, Expected Value/LOTUS, Dice, Markov Chains. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.