On Monday, you flip coins all day. You start flipping coins until you see the pattern Head, Tail, Head. Then you record the number of flips required, and start flipping again until you see that pattern, you record it, and start again. At the end of the day you average all of the numbers youâ€™ve recorded.
On Tuesday you do the EXACT same thing except you flip until you see the pattern Head, Tail, Tail.

Will Mondayâ€™s number be higher than Tuesdayâ€™s, equal to Tuesdayâ€™s, or lower than Tuesdayâ€™s?

Solution over the fold.

Tuesday’s will be lower.

The reason is as follows. Monday and Tuesday are the same until the sequence HT has been thrown. Monday and Tuesday have the same chance (50%) of a hit on the next throw. However, in the case of a miss they are different. On Monday a miss will be HTT and the next hit cannot begin until at least the throw after that. On Tuesday a miss will be HTH and the next hit can begin with the final H just thrown. The number of flips to get to a hit will be lower on Tuesday.

Calculating the exact value is hard, so I wrote a program instead. The result was: average 12.0 on Monday, 10.0 on Tuesday. I’m more confident about the logic than the accuracy of the computer program.