Bets are offered on a series of coin flips. Details have been added for this solution.

• The sequence HTT will appear before TTT. If it does, I give you $1. If not, you give me $4. (Good for you. Cost = .80, Expectation = .84)
• The sequence HHT will appear before TTT. If it does, I give you $1. If not, you give me $3. (Bad for you. Cost = .75, Expectation = .69)
• The sequence THH will appear before HHT. If it does, I give you $1. If not, you give me $2. (Good for you. Cost = .67, Expectation = .70)
• The sequence HTH will appear before THH. If it does, I give you $1. If not, you give me $1. (Good for you. Cost = .50, Expectation = .51)

The graphic below shows how the coin flip sequences compare. Code for generating these images is at the Coin Flips demo at the Wolfram Demonstrations Project. Recreational mathematicians can use this code in the newly released Wolfram Mathematica Home Edition. Previous puzzles are archived at the Math behind Numb3rs Season 5 page.