Six Dice Betting Game

A quick puzzle this week.

Your friend offers you a bet: Roll six dice, and if you get exactly four distinct numbers you win.

Question: would you accept this deal at even odds?

For example a roll of 1,2,3,6,3,6 would win but a 1,6,3,4,2,6 would be a loss.

Give it a few minutes thought …

Solution

There are many ways to solve this. One of these is with brute force. There are only 46,656 possible combinations the dice can land on (6  ×  6  ×  6  ×  6  ×  6  ×  6), and it’s just a couple of lines of code to write some nested loops and perform the check.

Here is a table showing the frequency count of every distinct combination of the spots.

Distinct Count	Frequency
1	6
2	930
3	10,800
4	23,400
5	10,800
6	720
TOTAL	46,656

As you can see, the total number of combinations with a distinct count of four is 23,400/46,656 which simplifies to 325/648, which is just a little more than 50:50, so it’s (just) slightly better than even odds. Yes, it’s a good bet to take!

Formal Solution

Solving with brute force might not be very elegant, but it’s simple to write. You could optimize a little from symmetry, but let’s apply logic and come up for a formal solution.

There are only two possible ways we can get a distinct count of four:

• Three of a kind, then three distinct singletons.

• Two (different) pairs, with two distinct singletons.

Triple

There are 6 possible values for what the triplet dice would be, and there are 6 choose 3 ways this triplet could appear. There are then 5 values that the first singleton could be, 4 values for the second, and 3 for the last singleton. This is 6  ×  ₆C₃  ×  5  ×  4  ×  3 = 7,200 possible combinations.

7,200/46,656 = 25/162.

Two Pair

With two pairs, there are ₆C₂ for what these numbers are. The first pair has ₆C₂ choices of where to go, and the second pair has ₄C₂ choices of position. There are then 4 possible combinations for the first singleton, and 3 possible for the last singleton.

This is ₆C₂  ×  ₆C₂  ×  ₄C₂  ×  4  ×  3 = 15  ×  15  ×  6  ×  4  ×  3 = 16,200 possible combinations.

16,200/46,656 = 25/72.

Total

Adding these together, we get (7,200+16,200)/46,656 = 23,400/46,656 = 325/648

This confirms the answer we obtained from brute force! (Approx 50.15%)