(See Doubling Cube Quiz )
Here are the answers:
- John and Jenny are playing a game of flipping pennies. Each flips one of their coins. If the coins are the same, John gets them both. If they are different Jenny gets them both. At the start of the game John has 8 pennies, Jenny has 12. They quit when someone has all of the money. What is the probability John will win?
Solution: This is a fair game so the average amount of money John will have at the end is the same as what he starts with. When he wins he has 20 pennies, so the probability he wins is
- Consider a flipping pennies game with 100 coins total. How many pennies should you have to offer a double? When should you accept?
Solution: Suppose the optimal threshold for offering a double is when you have a fraction p of the coins. At this point the two decisions, continue to play or give up, will have the same average payoff. By problem 1, the probability we will go from a probability of p to 1 before our probability slips to 1 − p (and the other person will redouble) is
2p − 1 p
for this makesp =
⋅ 1 +
2p − 1 p
⋅ (1 − p)
1 − p p
When we have probability p of winning, our expected value is 1 because the other player continuing or quitting has the same payoff. If they accept and we win, the payoff is 2. If they redouble, our payoff is the same if we drop, or −2. Thus the optimal p should satisfy1 = 2 ⋅
+ (−2) ⋅
2p − 1 p 1 − p p
Rearranging: 4 = 5p, or p = 0.8. You should double when your chance of winning is 80% or more. You should accept if your chance of winning is 20% or more.