Monday, June 11, 2007

The Sic-Bo Problem

So here's a puzzle:

Last night we went to the Melbourne Crown Casino, which is a labyrinth of neon madness the likes of which mankind was not meant to comprehend.

In the midst of the roulette, blackjack, pontoon and far-too-many poker machines, there is a little game that the Crown likes to call "Sic-Bo". The rules of Sic-Bo are simple - three six-sided dice are rolled, and you bet on what the result will be.

There are all sorts of complex bets you can make, but relevantly to our problem the bottom area of the betting table allows you to bet on the results of single dice. The numbers one through six are shown, and you can place your chips on any of these numbers.

If you bet on a number, and that number appears on any of the dice, the game pays 1 to 1. (That is, your $5 bet wins you a further $5.) If the number comes up on two dice, it pays 2 to 1 ($10 for your $5). And if all three dice show the number you bet on, it pays 12 to 1 ($5 yields $60).

This, I have to say, initially confused us. Those are good odds. I believe the phrase "money farm" was used. As we saw it, this was guaranteed returns, such that everything we knew and understood about casinos was turned upside down.

Here's how our reasoning went. If you place a $5 bet on each and every one of the six numbers, you couldn't lose money. In a case where all three of the dice show different numbers, you'd lose on half the numbers and win 1 to 1 on the others, thereby breaking even. Where you got a double and a single, that was the same total win as singles, so you were even. And on in the 1 in 36 occasions on which triples came up, you pocketed a profit. Rinse and repeat; increase your stake to make the money come in faster. There is no dice result on which the house doesn't pay out; cocked dice results in bets off but no loss to the player.

Using this logic we started doing all sorts of interesting maths that suggested we should immediately quit our jobs and just work the Sic-Bo tables 24/7, happily raking in buckets of cash while the other players looked on in an intoxicating mixture of awe and admiration.

So I have two questions. The first, which we have now solved, is in the case where we bet evenly on all six numbers, how does the house make its money? And the second question, which we haven't finished the numbers on yet, is does it make a difference if you instead just bet continually on a single number?


Arkem said...

This maths was done while I was very tired at the hotel so may be wrong. It is largely based around my foolings with ipython (the interactive interpreter for the Python programming language) I also had no Internet access at the time to check anything.

Out of the 216 possibilities of Sic-Bo:

A given number appears...
On one die: 75 times (+150)
On two dice: 15 times (+45 profit)
On three dice: 1 time (+13 profit)
On no dice: 125 times (0 profit)
Cost required is 216 bets
150 + 45 + 13 - 216 = -8

This suggests that if each possibility occurred once you would lose a total of 8 bets ($40 in the example given)

That's as far as I got before I decided against going back to the casino :)

Christopher Jam said...

I, too, was fiddling with Python last night (got as far as working out 10005**3 to get the 1,15,75,125 sequence, then some of the sums below). I then wrote the following comment, and was defeated by finding Google's terms of service too daunting before bedtime. (yes, I hadn't attached my rarely used blogger account my gmail yet). Trying again..

1) In the instance of a pair and a single, your winnings are the same as for three distinct, but you only get refunded on two of your six bets, hence you get $25 back for your $30 investment.

2) Betting on a single number repeatedly loses you on average 18.5 cents per bet. The probabilities of getting 3,2,1 or 0 occurrences of your number are 1/216, 3*(5**1)/216, 3*(5**2)/216 and 1*(5**3)/216 respectively, and the pay outs on a $5 bet are $65, $15, $10 and $0, giving a return of (1*65+15*15+75*10+125*0)/216, or $4.81481

GregT said...

Yep, that's the answer, all right. (*does a good impression of looking knowledgeable*)

Kremmen said...

I looked at the results of some of the other bets. All returned 83.3-87.5%. This particular bet returns 96.3%. If you are going to play Sic-Bo, it's definitely a good bet to make. Indeed, that'll lose you money more slowly than most bets on most casino games.

GregT said...

Which means the money I've allotted to lose lasts longer and I have more fun. Huzzah.