
The Sailor's GameCrown and Anchor is a traditional game that was popular with sailors of the Royal Navy in the 18th century. All it needs are three dice and a surface for placing bets which could either be a mat, or simply chalked out on a table or floor. The game was very quick to set up and play, and if necessary, the pieces could be snatched up and shoved away in a pocket if a disapproving officer approached. Although three normal dice can be used, the proper game dice have the four card suit symbols plus a crown and an anchor. The game itself is fast and simple, but the maths behind it is deceptive! 
How To Play
Any number of people can play against the Banker.
Suppose you bet $1 on all six symbols, and the dice land showing three different symbols, you'll receive 3 x $2 =$6 back. Nobody wins or loses! And if two matching dice pay double, and three dice pay treble, surely it all nicely evens out? Actually, it isn't quite as simple as that. 
The Banker's EdgeBe warned any professional banker would LOVE you to play Crown and Anchor! Let's look at the game from the Banker's point of view. For all these sums, we'll imagine there is $1 bet on each symbol for every roll of the dice. Therefore the banker will receive $6 every roll, but what would he expect to pay out?


The Banker makes his profit when a double or treble appears, but what proportion of singles/double/trebles can he expect? To work out these chances, we imagine the three dice landing one at a time. For a single (i.e. all the dice to land differently), the first dice can be anything. The chance of the next dice being different is 5/6, and the chance of the third dice being different from the first two is 4/6. Therefore the chances of the dice all landing differently are 1 x 5/6 x 4/6 = 20/36. The Banker pays back $6 when three different dice are thrown, so his expected payout is 6 x 20/36 = $3.33333 For a treble, the first dice can be anything. The second has a 1/6 chance of matching and the third also has a 1/6 chance of matching. The chance of a treble is 1 x 1/6 x 1/6 = 1/36. He pays back $4 on trebles so his expected payout is 4 x 1/36 = $0.11111 For a double the sums are slightly more complex but there's a nifty short cut. We know the singles chances are 20/36 and the treble chances are 1/36, and all the remaining throws must be doubles, so the chances are 120/36 1/36 = 15/36. He pays back $5 on doubles so so his expected payout is 5 x 15/36 = $2.083333 In total, for every $6 the banker receives, the expected payout is $3.33333 + $2.083333 + $0.11111 = $5.527777. Therefore his profit is $6  $5.52777 = $0.472223. The banker's percentage profit is 0.472223 / 6 = 7.87% So for every $100 bet, the banker can expect to make almost $8 in profit. This is a far better game for the banker than roulette, blackjack or most of the bets on craps. 
The Player's ChancesIf you put a single $1 bet on one symbol, what are your chances of winning? We'll use slightly different sums to the banker. Losing! The chances of your symbol not appearing on any of the three dice is 5/6 x 5/6 x 5/6 = 125/216. This works out at 57.87%. Roughly speaking, you'll lose 7 times out of 12. Single The chances of the first dice to land matching your symbol, and the next two not matching are 1/6 x 5/6 x 5/6 = 25/216. Of course it's the same chance for the second or third dice, so the total chance of matching one symbol is 3 x 25/216 = 75/216. This means you should match one symbol roughly once in every three throws. Double The chances of the first dice NOT matching your symbol, but the next two both matching are 5/6 x 1/6 x 1/6 = 5/216. Again, the nonmatching dice could appear in the second or third places, so the total ways of getting a double are 3 x 5/216 = 15/216. Treble The chances of the three dice all matching your symbol are 1/6 x 1/6 x 1/6 = 1/216. Just to check: the different chances added up are 125/216 + 75/216 + 15/216 + 1/216 = 216/216. It works! What are you likely to win? Let's say you bet $1 on the same symbol 216 times. In total that costs $216. Here are your expected payouts...
The player's percentage loss = 17/216 = 7.87% ... ...which of course is the same as the Banker's profit! 

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