I've got a probability question:
Given a 5-faced die (1,2,3,4,5),call it die $A$, each face has probability as follows:
$$\begin{array}{rrrrr} \text{Face} & 1 & 2 & 3 & 4 & 5 \\ \text{Prob} & 0.2 & 0.15 & 0.1 & 0.25 & 0.3 \end{array}$$
We roll this die three times and get $O = \{2,4,5\}$
Q1. What's the probability that we get this kind of outcome assuming that we are using die A
My solution is: $3!\cdot0.15\cdot0.25\cdot0.3$,
Q2. Given another 5-faced die $B$ and its probability distribution is as follows:
$$\begin{array}{rrrrr} \text{Face} & 1 & 2 & 3 & 4 & 5 \\ \text{Prob} & 0.1 & 0.2 & 0.3 & 0.25 & 0.15 \end{array}$$
Now, we have 2 dice, given that we do not know which die we rolled, but the outcome is $O = \{2,4,5\}$, whats the probability this die is die A?
Probability of die A, given outcome 245, equals (probability of die A) times (probability of 245 given die A), divided by the sum of [(probability of die A) times (probability of 245 given die A)] and [(probability of die B) times (probability of 245 given die B)].
Now you have calculated probability of 245 given die A, and you can similarly calculate probability of 245 given die B, but what you need to know is the a priori probability of die A and probability of die B. Perhaps you are meant to assume that these are both 1/2.
EDIT: The above concerns Q2. I believe the solution to Q1 in the original post is correct.