Gambler Board Game 654 Problem

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My parents bought me a board game called Gambler self-published by Charlie's Games from Nottingham. This is different from the Parker Brothers board game of the same name.

Gambler appears to have a cult following. I struggled to find any references to it on the Internet except for a reference in boardgamegeek.com. There is a reference to a home page, gambler-game.com but this appears to be just ads. There is reference to the Gambler World Championship in Nottingham in 2003. I found someone blogging about the game.

One of the "gambles" in Gambler is to try and roll the numbers 6, 5, 4 using five dice. You have up to three rolls to make the 6, 5, 4. You can keep any number of dice with the additional rule that you can always keep a 6, you can only keep the 5 if you already have a 6. You can keep a 4 only if you have a 5 (and also the 6, i.e. you've won).

This option is available if you have to pick your own gamble during the game - another option is tossing a coin - clearly a 50-50 proposition so should you ever choose the 654 game?

This is also a good maths problem - what are the odds of making a 6, 5 and 4 with 5 dice in three rolls?

I do not know of any magic formula, so....

Solution

First Throw

For the first throw, all five dice are rolled. There are four cases:

There are 6^5 = 6 * 6 * 6 * 6 * 6 = 7,776 possible outcomes of rolling 5 dice.

A will occur 1,230 times. (This is where you get to do the maths!)

B will occur 1,320 times.

C will occur 2,101 times.

D will occur 5^5 = 3,125 times.

Second Throw

For the second throw, we only need to consider cases B, C, D (A is a winner).

For B, we need to throw a 4 with two throws of three dice. There are three cases:

Double checking, 19656 + 11375 + 15625 = 46656.

For C, we are throwing four dice. There are 6^4 = 1,296 different outcomes of throwing 4 dice. There are three cases:

For D (nothing on first throw), we are back to the same odds as when we started, only we now have just two rolls. We can then create the following cases (based on those above): DA, DB1, DB23, DC1, DC23, DDA, DDD

The worst case will be having to throw all five dice three times. We use this as the lowest common denominator (6^15)=470184984576.

Putting this all together
Odds of throwing a 6, 5, 4 with 5 dice in 3 rolls
Case Roll/Odds Number
numerator/denominator
Win # of winning
cases/470184984576
# of losing
cases/470184984576
Notes
1 2 3
A 1230/7776 1 1 1230/7776 Y 74373396480 0 654 in first roll
B1 1320/7776 91/216 1 120120/1679616 Y 33625912320 0 65 in first roll, 4 in second roll
B2 1320/7776 125/216 91/216 15015000/362797056 Y 19459440000 0 65 in first roll, nothing in second roll, 4 in third roll
B3 1320/7776 125/216 125/216 20625000/362797056 N 0 26730000000 65 in first roll, nothing in second or third roll
C1 2101/7776 302/1296 1 634502/10077696 Y 29603325312 0 6 in first roll, 54 in second roll
C2B1 2101/7776 369/1296 91/126 70549479/2176782336 Y 15238687464 0 6 in first roll, 5 in second roll, 4 in third roll
C2B23 2101/7776 369/1296 125/216 96908625/2176782336 N 0 20932263000 6 in first roll, 5 in second roll, nothing in third roll
C3C1 2101/7776 625/1296 302/1296 396563750/13060694016 Y 14276295000 0 6 in first roll, nothing in second roll, 54 in third roll
C3C23 2101/7776 625/1296 994/1296 1305246250/13060694016 N 0 46988865000 6 in first roll, nothing in second roll, no 54 in third roll
DA 3125/7776 1230/7776 1 3843750/60466176 Y 29889000000 0 Nothing in first roll, 654 in second roll
DB1 3125/7776 1320/7776 91/216 375375000/13060694016 Y 13513500000 0 Nothing in first roll, 65 on second roll, 4 on third roll
DB23 3125/7776 1320/7776 125/216 515625000/13060694016 N 0 18562500000 Nothing on first roll, 65 on second roll, nothing on third roll
DC1 3125/7776 2101/7776 302/1296 1982818750/78364164096 Y 11896912500 0 Nothing on first roll, 6 on second roll, 54 on third roll
DC23 3125/7776 2101/7776 994/1296 6526231250/78364164096 N 0 39157387500 Nothing on first roll, 6 on second roll, nothing on third roll
DDA 3125/7776 3125/7776 1230/7776 12011718750/470184984576 Y 12011718750 0 Nothing on first two rolls, 654 on third roll
DDD 3125/7776 3125/7776 6546/7776 63925781250/470184984576 N 0 63925781250 Nothing on first two rolls, no 654 on third roll
Totals: 253,888,187,826 216,296,796,750  
%: 53.9975108  

Therefore, approximately 54% of the time you will roll a 654 with 5 dice.

So, if you are in the World Championship, you land on Any Gamble, the odds of Trap Flier aren't that good, the cards are only 50-50, the coin is only 50-50 consider the 654 to win the championship!