You find yourself in Las Vegas. One of the casino games involves rolling two 6-s

Question

You find yourself in Las Vegas. One of the casino games involves rolling two 6-sided dice and betting on the outcome – the sum of the two dice. For example, you can bet that the sum of the two dice will be 8 (2 and 6, 3and 5, etc.). The price to bet, the probabilities, and the winning payoffs are as follows:
Sum to Two 6-sided Dice 2 3 4 5 6 7 8 9 10 11 12
Price to Play $1.00 $1.00 $1.00 $1.00 $1.00 $1.00 $1.00 $1.00 $1.00 $1.00 $1.00
Probability 0.02778 0.05556 0.08333 0.11111 0.13889 0.16667 0.13889 0.11111 0.08333 0.05556 0.02778
Winning Payoff $36.00 $18.00 $12.00 $9.00 $7.20 $6.00 $7.20 $9.00 $12.00 $18.00 $36.00

You have discovered that one of the dice was manufactured incorrectly. Normally, each number on a given dice has a 1/6 chance (.1667) of occurring. However, the actual probabilities for the two dice are as follows:

1: Dice 1: 0.17 Dice 2: 0.1667

2: Dice1: 0.17 Dice 2: 0.1667

3: Dice 1: 0.17 Dice 2: 0.1667

4: Dice 1: 0.17   Dice 2: 0.1667

5: Dice 1: 0.17 Dice 2: 0.1667

6: Dice 1: 0.15   Dice 2: 0.1667

You construct an arbitrage opportunity by going Long on 6 and Short on 8. Calculate the expected payoff and discuss any risk involved with this strategy, if present.

Explanation / Answer

Sum of points

of 2 dice

Probability for

unbiased dice

Probability for

given biased dice

For biased case expected payoff being Long on 6 and short on 8, will be = (0.141695×7.2)+(0.163366×6)+(0.135027×7.2) = $2.9725944

For unbiased case expected payoff being Long on 6 and short on 8, will be = (0.13889×7.2)+(0.16667×6)+(0.13889×7.2) = $3.000036

So, there is a risk involved with this strategy which is the amount $0.0274416 will be less from the actual expected amount.

Sum of points

of 2 dice

Probability for

unbiased dice

Probability for

given biased dice

Winning payoff 2 0.02778 0.028339 $36 3 0.05556 0.056678 $18 4 0.08333 0.085017 $12 5 0.11111 0.113356 $9 6 0.13889 0.141695 $7.2 7 0.16667 0.163366 $6 8 0.13889 0.135027 $7.2 9 0.11111 0.106688 $9 10 0.08333 0.078349 $12 11 0.05556 0.050010 $18 12 0.02778 0.025005 $36

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