Sanderson M. Smith

TWO DICE LUBER

(Casino game constructed by Cate students Brad, Emily, Anne, and Aaron)

Here is the game description as written by the inventors:

The player puts down a bet of two (2) dollars. The payout will be specific to what the bet, or bets for that matter, were placed on. If the player wins, he/she gets back the two dollar bet, or however much was originally bet. The player has the option of betting on the outcome of the role of two dice, excluding outcomes of 6 and 8. The players can also bet on other options: multiples of 3, multiples of 4 (all multiples inclusive), even numbers, and/or odd numbers. Again, 6 and 8 pay nothing and are just ignored. In essence, this game is similar to roulette, only differing with the probabilities of how certain numbers can appear on two dice.

The payouts are as follows:

• roll a 2 (snake eyes)- 35:1
• roll a 3 ( 2 and 1)- 17:1
• roll a 4 (2 and 2, 3 and 1)- 11:1
• roll a 5 (2 and 3, 4 and 1)- 8:1
• roll a 7 (3 and 4, 5 and 2, 6 and 1)- 5:1
• roll a 9 (4 and 5, 6 and 3)- 8:1
• roll a 10 (5 and 5, 6 and 4)- 11:1
• roll an 11 (6 and 5)- 17:1
• roll a 12 (6 and 6)- 35:1
• multiple of 3 (3,9,12)- 5:1
• multiple of 4 (4,12)- 8:1
• even number (2,4,10,12)- 4:1
• odd number (3,5,7,9,11)- 1:1

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Now, taking the rules literally, the totals 6 and 8 are not part of this game. The rules, as written, say "6 and 8 pay nothing and are just ignored." In other words, you don't lose if you roll a 6 or 8, an interpretation that, as we will see, makes the game very favorable for the player.

Without the 6 and 8 totals, there are only 26 meaningful outcomes in this game. (See diagram at the right.)

An analysis of this game appears below. The analysis is based on what can be expected per \$1 spent to play Two Dice Luber.

 1 2 3 4 5 6 1 2 3 4 5 7 2 3 4 5 7 3 4 5 7 9 4 5 7 9 10 5 7 9 10 11 6 7 9 10 11 12

 EXPECTATION IN 26 GAMES (PER \$1 SPENT) Payoff EVENT # games player wins # games house wins Dollars player wins Dollars casino wins Player gain (26 games) Player gain per game 35:1 total=2 1 25 \$35 \$25 \$10 \$.3846 17:1 total=3 2 24 \$34 \$24 \$10 \$.3846 11:1 total=4 3 23 \$33 \$23 \$10 \$.3846 8:1 total=5 4 22 \$32 \$22 \$10 \$.3846 5:1 total=7 6 20 \$30 \$20 \$10 \$.3846 8:1 total=9 4 22 \$32 \$22 \$10 \$.3846 11:1 total=10 3 23 \$33 \$23 \$10 \$.3846 17:1 total=11 2 24 \$34 \$24 \$10 \$.3846 35:1 total=12 1 25 \$35 \$25 \$10 \$.3846 5:1 multiple of three (3,9,12) 7 19 \$35 \$19 \$16 \$.6154 8:1 multiple of four (4,12) 4 22 \$32 \$22 \$10 \$.3846 4:1 even (2,4,10,12) 8 18 \$32 \$18 \$14 \$.5384 1:1 odd (3,5,7,9,11) 18 8 \$18 \$8 \$10 \$.3846

So, if you had an opportunity to play Two Dice Luber in a casino with the indicated payouts, your best option would be to always bet on a multiple of 3 (which wouldn't include the ignored total of 6). For each dollar "invested" on this event in Two Dice Luber, you would expect to gain about 62 cents. From a formal probability standpoint, your expected return on this event is

(\$6)(7/26) + (\$0)(19/26) = \$1.6154

Remember, if you win, you get your dollar back, plus another \$5. When betting on the event "multiple of 3," your expectation is to win about \$1.62 for each dollar you "invest," making your expected gain to be 62 cents a game.

Two Dice Luber, with the indicated payoffs, would be very popular with gamblers. But, don't expect to find the game in a casino.