# Bayes' rule and probability

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?

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Note: not "Baye" ... named for Rev. Thomas Bayes. The Baye Rule is as nonexistent as the Stoke Theorem. Write "Bayes' Rule" or "Bayes's Rule" or something. – GEdgar Sep 14 '12 at 0:30
MAYBE HE ROLLS B AND GETS THE [2.4.5] OUTCOME WITH THREE ROLLS. ASSUMING YOU KNOW THE MULTINOMIAL DISTRIBUTION FOR A WHAT IS THE P VALUE OF THE TEST THAT THE DIE USED IS A. tHE NULL IS THAT IT IS A AND THE ALTERNATIVE IS NOT A. IF THIS FORMULATION IS CORRECT THE DISTRIBUTION FOR B IS IRRELEVANT. @LouisTan Do I have the correct interpretation? – Michael Chernick Sep 14 '12 at 2:55
@Michael, please stop SHOUTING. – Gerry Myerson Sep 17 '12 at 6:10
@GerryMyerson Occasional capitalization for emphasis is okay I think. I don't like to think about it as shouting. Please try to spell Louis' name correctly. – Michael Chernick Sep 17 '12 at 11:34
@Michael, please, there is a difference between OCCASIONAL capitalization and a whole paragraph of it. – Gerry Myerson Sep 17 '12 at 12:05

## 1 Answer

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.

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+1 This seems to be a reasonable interpetation if Bayes' rule is to be appied. It assume the multinomial distributions are known and that it is not known if A or B is chosen but the decision was made by say flipping a fair coin with A chosen if say it lands heads. – Michael Chernick Sep 14 '12 at 3:03