## How to Win at Coin Flipping

Let's flip a coin, over and over. Beforehand, the players each pick a sequence of flips. The sequence that occurs first wins. With HH vs. TH, HH will win if the first two flips are HH and will lose if any of those flips are tails. HH vs. TH has a 1/4 vs. 3/4 possibility of winning.
Phrased a different way: Suppose I offer a bet on a series of coin flips. One of these bets would be bad for you. Which one? The odds of the event occurring are given at the end. (The second bet is the bad bet to take.)
HTT appears before TTT. If it does, I give you $1. If not, you give me $4. (7/8)
HHT appears before TTT. If it does, I give you $1. If not, you give me $3. (7/10)
THH appears before HHT. If it does, I give you $1. If not, you give me $2. (3/4)
HTH appears before THH. If it does, I give you $1. If not, you give me $1. (1/2)
This is the strange world of Penney's game. Here is a table of odds and facts for various 3-flip games. Calculating these odds can be both tedious and mathematically demanding—a natural job for

*Mathematica*or*Mathematica Home Edition*.