Probability Dispute

You and your friend Bob are disputing the probability of getting a 7 by rolling 2 standard dice. Your claim is that the probability of getting 2 heads from rolling 2 dice is 6 out of 36, or 1 in 6, chance of rolling a 7. Your friend claims that the probability should 3 out of 21, or 1 in 7, because the order of the dice is irrelevant, whether it be rolling a {3,4} or {4,3} should be treated equally.

Frustrated by trying to convince your friend Bob about his error logically, you instead set up a simulation to test this. You and your friend Bob are both gamblers so you decide to settle this dispute with a few wagers. Bob will give 6 to 1 odds that rolling 2 dice will yield a 7, a $1 bet could win an additional $6 (on top of getting back your $1 wagered), for a bet for the sum of 2 dice being a 7. Bob feels that these odds reflect a fair game and at the end of this simulation that both players, on average, should end up with the same amount of money when started. You feel that this game slightly favors you and that you will end up with more money than initially.

You have $18 that you are willing to spend on this dispute. Based off the Kelly Criterion, what fraction of your money should you bet each time in order to optimize your wealth from this game?

*Note the Kelly Criterion states: * In probability theory and intertemporal portfolio choice, the Kelly criterion, Kelly strategy, Kelly formula, or Kelly bet is a formula used to determine the optimal size of a series of bets in order to maximize wealth.

where: $f^{*} = \frac{bp - q}{b} = \frac{bp - (1 - p)}{b} = \frac{p(b + 1) - 1}{b}$

f * is the fraction of the current bankroll to wager, i.e. how much to bet;

b is the net odds received on the wager ("b to 1"); that is, you could win $b (on top of getting back your $1 wagered) for a $1 bet

p is the probability of winning;

q is the probability of losing, which is 1 − p.