A release-checked medium problem for training Conditional Expectation, Combinations and Permutations, Coins, Expected Value/LOTUS.
Question
Varun has $4$ fair coins. He flips all $4$ at once and notes the parity of each. After seeing the outcomes, he may flip any pair of tails again. Varun may not flip a single coin without flipping another. He can iterate this process as many times as he would like, as long as he has less than $3$ heads obtained (otherwise, no pair of tails exists, and the game ends). Find the expected number of flips that Varun performs. Note that the initial $4$ flips are still flips, and each pair counts as $2$ flips!
Practice focus
This Probability problem is tagged Conditional Expectation, Combinations and Permutations, Coins, Expected Value/LOTUS. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.