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?
How does the casino make money on this game? Do they sell drinks or something?
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.
Crikey, Stu, this is difficult. I'm not sure the rules are completely clear; an assumption I've made is that if you get, say, Bar Bar Not-Bar, then respin the Not-Bar, and get a Cherry, then you don't get another respin.
The answer is going to depend on how many cherries you hold under which circumstances. There are three ways to play the game if you get two cherries (hold 2, hold 1, hold 0) and two ways to play the game if you get one cherry (hold 1, hold 0). Holding 1 cherry in either case makes the maths complicated – more complicated than I know how to do off the top of my head, though I may be able to get someone to teach me how to do it.
However, I've been able to calculate the expected return if you hold two cherries when you get two cherries and if you hold no cherries when you get one cherry. (I don't claim this is the optimal way to play the game.) I think your expected return is (7*pot – 183) / 312, so the game is in your favour if the pot contains more than 183/7 pounds (i.e. 27 pounds or more) before you pay the pound to play.
You can see some of my working <a href="https://docs.google.com/spreadsheet/ccc?key=0Au6hWFl_m0XJdHZldjJiaUNyQWlBVFprT0l3SWFzNUE&hl=en_US">in this Google Docs spreadsheet</a>, – this is case A for 2 cherries and case B for 1 cherry – but I've added the figures up on paper at the end.
I've been able to improve on that. If you never hold any cherries and you always respin everything (case C for 2 cherries and case B for 1 cherry) then I think your expected return is (7*pot – 169) / 282, so the game is in your favour if the pot contains more than 169/7 pounds (i.e. 25 pounds or more) before you pay the pound to play. And if it is, then this is always a better strategy than holding two cherries if you get them both.
I have a gut feeling that respinning everything whenever you get cherries is probably going to be the best play simply because the prize for three cherries is so poor, but I haven't been able to show this for certain.
You like writing, Stu, and I like reading what you write. I like solving puzzles, and you may still want to know what the answer to this puzzle is. (Not that I've got the answer for certain yet, but I do think I've answered this for two small cases.) It seems like the least I can do under the circumstances.
By the way, I'm the bloke who gave you that link to the site that had the Slitherlink on the funny grid that other month – since then they've had a Slitherlink which was on a grid of triangles and hexagons. If they do funny slitherlinks again do you need me to tell you about it?
I always need to hear about funny slitherlinks.