# Avoid the Scam

For this last set of questions, we're going to analyze two dice games; both have been played in real casinos.

For the first game, Under and Over, there are three possible bets:

The player picks a square and bets however much they want, and then rolls two dice. They win on Under 7 if they roll 2, 3, 4, 5, or 6. They win on Over 7 if they roll 8, 9, 10, 11, or 12. On 7, they only win if they roll exactly 7.

There is a $\frac{3}{36} = \frac{1}{6}$ probability of rolling a 7 on a pair (count on the grid below). What is true about the probabilities of Over and Under?

# Avoid the Scam

In casino bets, the most likely to win isn't necessarily the best, because bets don't all pay equally:

With Under and Over, a bet of $100 on Under or Over wins back$100; a bet of $100 on 7 wins$400.

You can use the expected value to compare the outcomes; which square is best to bet on?

# Avoid the Scam

As found from the solution in the last question, assuming a bet of $100 per turn, the player has an expected value of$83.33 per turn no matter which square they bet on.

# Avoid the Scam

The previous calculation led to finding betting $100 every turn leads to an expected value to the player of$97.22.

What is the house advantage? (That is, in the long run, what percent of each dollar spent on the game does the casino get?)

# Avoid the Scam

To compare our two games, we had a house advantage of 16.67% vs. one of 2.78%. It means, in essence, Over and Under will earn money 6 times faster for the casino than Simplified Craps. (2.78% is generally considered reasonable by gamblers, although there are games where the house advantage is even smaller.)

With both these games, there weren't any choices that mattered, but some casino games involve a wide variety of bets that not only need to be analyzed separately but can change the house advantage. This course will consider situations in Blackjack, Craps, and Poker, and get you to practice in finding the best way to win!

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