So in this question, I understand the first bit on combinations for rolling a 5, but I don’t understand why 1/27 was used for the second bit on the either 1 or 2.
Why does Greg use 3 options for 5, and only 1 option for either 1 or 2? I was expecting to use 3 options for 5, and then 8 options for either 1 or 2 because the order can be arranged in 8 ways. Thus I got a 5/243 but Greg’s answer is 15/5832.
Help please.
Let (X_1, X_2, X_3) denote the outcomes of the last three rolls. For example, the outcome (1,2,1) would mean that the fourth roll was a 1, the fifth was a 2, and the sixth was a 1.
For each of these rolls, we are looking for two specific outcomes (either a 1 or a 2) from \{1,2,3,4,5,6\}. Thus, the probability for each roll is \frac 26. Since there are three independent rolls, we get the following:
\frac 26 \cdot \frac 26 \cdot \frac 26 = \left( \frac 13 \right)^3 = \frac {1}{27}
