## Monday, 23 June 2014

### Gambling and Expected Value: Crown and Anchor

In this post on Gambling and Expected Value, we look at the game Crown and Anchor.
Click here to find similar posts on other lotteries and games of chance.

### How the Game Works

The game Crown and Anchor is a dice game that historically was often played by British sailors. Instead of ordinary dice, each face has one of six symbols: crown, anchor, spade, diamond, club, and heart.

 A Crown and Anchor mat with dice.

Players can bet on one or more symbols coming up and then roll three dice. If the symbol you bet on comes up, you win. The payout depends on how many of the dice are showing the symbol.

There is also a variant of the game played with a wheel instead of dice. The wheel is divided into 216 segments, where each segment shows one the 216 possible ways to choose from the six symbols three times. The banker in this case spins the wheel and the winner is determined based on which segment is showing when the wheel stops.

### Probabilities and Prizes

If the winning symbol appears on one of the three dice, the payout is 1:1 on the wager. If two of the three dice are showing the winning symbol, it's a 2:1 payout. Likewise, if all three dice are showing the symbol, it's a 3:1 payout. So there are four possible outcomes for a wager (loss, 1:1 payout, 2:1 payout, and 3:1 payout). Each die has 6 symbols, so there are 6 * 6 * 6 = 216 possible combinations of symbols when three dice are rolled. The probability of losing is 5/6 * 5/6 * 5/6 = 125/216. That leaves 91 possible ways to win (approximately a 42% chance of winning something). Only one of those combinations would be all three symbols matching the wager. Therefore, there's a 1 in 216 chance of winning the 3:1 payout. There are 15 possible ways to get exactly two matching symbols, so there is a 15 in 216 = 5 in 72 chance of winning the 2:1 payout. Finally, there are 75 ways to get exactly one matching symbol, corresponding to a 75 in 216 = 25 in 72 chance of winning the 1:1 payout.

### Expected Value

The expected value of Crown and Anchor is about -\$0.079 per dollar wagered, which was calculated as follows:

This means that, on average, you lose nearly 8 cents of every dollar wagered. A lot better than lotteries or Monte or 50/50 draws, allowing you to play longer with your money. It's much more comparable to casino games, though there are other options with even better expected value.

1. I don't have a massive amount of experience with statistics, but that second last equation doesn't seem to take into account the fact that you bet \$1 regardless of whether you win or lose. To my mind it should be:
EV=1/216*3+5/72*2+25/72-1

1. If you win, your wagered dollar remains on the table and the payout is added to it. The alternate way of expressing the expected value would be:
EV = (1/216)(3+1) + (5/72)(2+1) + (25/72)(1+1) - 1 = -17/216.

2. This comment has been removed by the author.

3. Because there are 6 symbols and each has as group of 3 on the wheel would that not make the odds of a 3:1 payout 6 in 216 or 1:36? Not 1:216?

1. There is a 1:36 chance that the wheel lands on one of the spots that pays out 3:1, and a 1:6 chance that you happened to pick the right symbol. 1/36 * 1/6 = 1/216.

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