Q:  What is higher: the probability of rolling a sum of 8 when two dice are rolled or when 3 dice are rolled?

A:  Let’s start with the first part… What is the probability of rolling an 8 when two dice are rolled?  Well, when you roll two dice there are 36 possible outcomes (6 outcomes on 1 die * 6 outcomes on the other die = 36 possible outcome).

Now, how many of these outcomes equal 8?  The pairs that are possible on two dice that make 8 are:

(2, 6) and (6, 2)

(3, 5) and (5, 3)

(4, 4)

So, there are 5 possible ways to get an “8”.  The probability of rolling an 8 on two dice is 5/36.

Now, let’s consider rolling 3 dice.  There are 6 * 6 * 6 = 216 possible outcomes from 3 dice.  How many of these outcomes make “8”?  For fun, I am going to list each permutation followed by each combination (the same numbers, just scrambled):

(1, 1, 6) and (1, 6, 1) & (6, 1, 1)

(1, 2, 5) and (1, 5, 2) & (2, 5, 1) & (2, 1, 5) & (5, 1, 2) & (5, 2, 1)

(1, 3, 4) and (1, 4, 3) & (3, 1, 4) & (3, 4, 1) & (4, 1, 3) & (4, 3, 1)

(2, 2, 4) and (2, 4, 2) & (4, 2, 2)

(2, 3, 3) and (2, 3, 2) & (3, 2, 2)

Note:  It is very important to do this systematically so that you don’t repeat any accidentally AND so you don’t forget any.  Can you follow the way I did it?

Now, count them all up… There are 21 possible ways to roll an “8”.  Therefore, the probability of rolling an 8 on three dice is 21/216.

Now compare?  Who is larger? 21/216 or 5/8….? WAAA… 5/8 is MUCH larger.

Therefore, it is much more likely to roll an 8 on two dice than it is to roll an 8 on three dice!