﻿ Chuck-a-Luck

# Chuck-a-Luck

 There is a dice game you might have seen in casinos called Chuck-a-Luck (it is sometimes called Birdcage). The game is played by attempting to guess the outcome of three dice that are captive in a cage.
 Bets are placed, then the cage containing the dice is inverted. When the dice come to rest their top faces are examined and any winning bets paid out. This article takes a look at the odds of the game. The game that is played in America is a variant of an Asian game called Sic-Bo. There are a variety of bets possible but the one we’ll look at first is the single number bet.

## Single Number Bet

To place a single number bet, the player selects one of the dice values 1-6, then the dice are rolled.

After the roll, if any one of the dice match, the player wins, earning even odds. If any two of the dice match the selected number, the player wins 2:1 odds on his bet. If all three of the dice match the selected number, the player wins 10:1 on his bet.

Match any one dice Match two dice Match all three dice
Payout 1:1 Payout 2:1 Payout 10:1

To work out the expected payout of the game we need to examine the odds of of each of the three possible outcomes.

• There is a 1/216 chance that all three dice match the selected number (1/6 × 1/6 × 1/6).

• There is a 15/216 chance that two out of the three dice match (3 × 1/6 × 1/6 × 5/6).

• There is a 75/216 chance that just one of the three dice match (3 × 1/6 × 5/6 × 5/6).

• There is a 125/216 chance that none of the dice match (5/6 × 5/6 × 5/6).

The expected outcome for a \$1 bet is:

Ex = (10 × 1/216) + (2 × 15/216) + (1 × 75/216) − (1 × 125/216)

Ex = − 10/216 ≈ −0.0463

You can see why casinos love this game, the odds are clearly in their favour. On each round of the game, the player will get an expected 95.37% back.

Not all casinos are even as generous as this. Some don't even pay 10:1 on matching all three dice, some pay only as little as 3:1 odds. Depending on the payout for matching three of a kind, the expected outcome changes.

30:14.630%
29:14.167%
28:13.704%
27:13.241%
26:12.778%
25:12.315%
24:11.852%
23:11.389%
22:10.926%
21:10.463%
20:10.000%
19:1-0.463%
18:1-0.926%
17:1-1.389%
16:1-1.852%
15:1-2.315%
14:1-2.778%
13:1-3.241%
12:1-3.704%
11:1-4.167%
10:1-4.630%
9:1-5.093%
8:1-5.556%
7:1-6.019%
6:1-6.481%
5:1-6.944%
4:1-7.407%
3:1-7.870%

To the left is a table showing the advantage to the player for a variety of odds paid out for rolling three of a kind.

For all odds below 20:1 the casino has the advantage. With payouts at 20:1, the game is a wash, and if the game were played with payouts on three mathching greater than 20:1, then the player would have an expected advantage.

## Any Triple

This is another kind of bet.

A bet placed here will win if all three dice match (one does not need to specify which number). This will happen, on average, one time in 36. The casino pays 30:1 on this bet. The expected outcome is therefore:

Ex = (30 × 1/36) − (1 × 35/36)

Ex = − 5/36 ≈ −0.1389

The casino has the advantage. To make the bet neutral, the casino would need to pay 35:1 on this wager.

## Big Bet

Another kind of bet is the Big Bet. This pays out if the dice roll a total of 11 or higher, but excluding three of a kind. This means that rolls of {6,6,6}, {5,5,5}, and {4,4,4} are not winning rolls.

Of the 216 possible rolls, there are 108 rolls that total 11 or higher. From this we have to subtract the three triples, leaving 105 winning rolls. The casino pays even odds on this bet.

Ex = (1 × 105/216) − (1 × 111/216)

Ex = − 6/216 ≈ −0.0278

As before, this is in the casino's favour. If we did not have to remove the three triples, it would be a wash.

## Small Bet

A symmetrical bet is the Small Bet. On this bet the casino pays out even odds on rolls of 10 or lower (again, excluding triples).

As above there are 105 winning rolls, and the expected outcome and house advtange are the same.

## More Bets

Some casinos also offer other spreads of bets: Odd rolls, Even rolls, Field Bets (rolls landing in certain bands of numbers) … Whatever the offered bets, you can be sure that they will all be in the casino's favour!