## Math(s) rock(s)

So here’s some mental exercise for this morning. A casino introduces a new table game. It involves rolling three dice with six fruit-machine symbols on them instead of the normal pips. At the start of a game the house puts **£10** in the pot and players pay **£1** into the pot to roll the three dice.

The payouts and rules run as follows:

**Three Bars – player scoops the entire pot**.

(the house then starts a new pot with £10)

**Three Bells or three Plums – player wins £10.**

**Three Oranges or three Lemons – player wins £5.**

**Three Cherries – player wins £3.**

There are two further rules. If a player rolls two of the same symbol, they may pay another £1 into the pot and roll the third dice one more time to try to complete the set. And if the player rolls one or two cherries, they may “respin” by re-rolling however many dice they wish, for free.

(If there are two cherries the player can have two respins, but must “hold” one of the cherries on the first respin. If any re-rolled dice come up cherries, they grant another respin, but a cherry can only grant one respin without being re-rolled.)

The question, then, is this: **how good (or bad) are the odds in this game for the player?** Here are a few figures to help.

– There are **216** possible outcomes of three six-sided dice.

– **6** of these are instant wins (straight triples), with odds of **35/1** against.

– **66** outcomes are doubles with no cherry. Paying for the second roll produces total odds of **27/1** on rolling three of a kind via a paid respin.

– **14** outcomes feature two cherries.

– **10** outcomes are doubles with one cherry, giving total odds of **13.5/1** on rolling three of a kind this way.

– **78** outcomes feature a single cherry with two other different fruits

The full list of possible outcomes from three dice can be found here.

Are you going to bust the casino or lose your shirt, viewers?

February 2, 2010 at 12:58 pm

How does the casino make money on this game? Do they sell drinks or something?

February 2, 2010 at 1:18 pm

That’s a good question. For the sake of argument let’s say they reset the pot every hour or every 100 throws or something.