Here is a brainstorming session on coin flipping. It involves a game of pattern choosing. You have to choose a pattern which have better probability to occur.

- Flip a fair coin twice. What is the probability that you get two heads (HH)? What is the probability that you get heads followed by tails (HT)? Are these probabilities the same?

*Yes, sure they are. No confusion yet.*

- Flip a fair coin repeatedly until you get two heads in a row (HH). On average, how many flips should this take? What if we flip until we get heads followed by tails (HT)? Are the answers the same?

*Based on the previous problem, most folks assume that the answer is Yes, but the answer is No.*

- Let’s play a game: we flip a coin repeatedly until either HH emerges (I win) or HT emerges (you win). Is the game fair?

*Based on the previous problem, most folks assume that the answer is No, but the answer is Yes.*

- We play the let’s-flip-a-coin-until-a
-pattern-emerges game. You pick HHT as your pattern, I pick THH. We flip a fair coin repeatedly until we get heads-heads-tails in a row (you win) or tails-heads-heads in a row (I win). Is the game fair?

*Based on symmetry, most folks assume that the answer is Yes, but the answer is No. I’m winning 75% of the games*

- Seeing as THH is a better pattern, you request to pick it as your pattern. I graciously agree, and switch to TTH. I keep beating you most of the time. You switch to my TTH. I switch to HTT. I keep beating you. You switch to HTT. I switch to… HHT, your original losing pattern. Who’s winning now?

*Most folks are by now expecting trouble, and guess correctly even though they feel it’s impossible. Indeed, HHT is better than HTT, which is better than TTH, which is better than THH, which is better than HHT.*

This delightful result is known asĀ Penney’s game