My friends and I play a dice game with three dice. One of the winning combinations is 4-5-6. Which can be made with any of the dice as long as the sequence comes via any of the three dice. I figured that those odds were more common to get a 4-5-6, than to get triple of one number, say 5-5-5 for example.
I’m no math wiz but confident that 4-5-6 has better odds than any 3 of a kind (5-5-5 for this example); could someone explain the odds of each sequence as they don’t seem equal to me?? (The three dice are all standard 6-sided numbered 1-6)
There is only 1 way to get a 5-5-5 sequence, as you have to get a 5 on each dice to get the 5-5-5 sequence. There are $6^3=216$ possible sequences with 3 dice, so the odds of getting a 5-5-5 sequence is 1 in 216, or about 0.46% (this goes for any 3-of-a-kind sequence). However, for the 4-5-6 sequence you have 3 possibilities for the first dice (a 4, a 5 or a 6). For each possibility on the first dice you have 2 possibilities for the second dice (one of the other 2 numbers in the sequence that did not appear on the first dice), and then you have 1 possibility for the final dice (the final remaining number). Therefore, there are $3\times 2\times 1=6$ possible ways of getting the 4-5-6 sequence with 3 dice, so the odds of getting that sequence is 6 in 216, or about 2.8% (this goes for any seqeunce of 3 distinct numbers).