Event $A$ represents my belief that a three-sided dice is fair, i.e: PMF of $P(A)$ is given by:
$f_A(w_k) = \frac{1}{3}$
where $w$ represents a single distinguishable outcome and $k$ represents one of the face of the die.
Some new evidence comes along (represented by event $B$) that suggests that my dice is not fair.
Given that the distirbution of P(B) is:
$w_1=\frac{1}{3}$, $w_2=\frac{1}{6}$, $w_3=\frac{1}{2}$
Calculate the actual probabilties of my dice.
Using Bayes:
$P(A | B) = \frac{P(A, B)}{P(B)}$
Step 1: Calculate likelihood:
$P(A,B) = P(A \cap B)$
Assuming A and B are indepdent then:
$P(A=w_1 \cap B=w_k) = \frac{1}{3}*\frac{1}{3}+\frac{1}{3}*\frac{1}{6} +\frac{1}{3}*\frac{1}{2} = \frac{1}{3}$
$P(A=w_2 \cap B=w_k) = \frac{1}{3}*\frac{1}{3}+\frac{1}{3}*\frac{1}{6} +\frac{1}{3}*\frac{1}{2} = \frac{1}{3}$
$P(A=w_3 \cap B=w_k) = \frac{1}{3}*\frac{1}{3}+\frac{1}{3}*\frac{1}{6} +\frac{1}{3}*\frac{1}{2} = \frac{1}{3}$
This implies that $P(A \cap B)= [w_1= \frac{1}{3},w_2= \frac{1}{3},3=w_3= \frac{1}{3}]$
Step 2: Calculate Marginal likelihood:
$P(B) = [w_1=\frac{1}{3}, w_2=\frac{1}{3}, w_3=\frac{1}{3}]$
This implies that $P(A | B) = [1,1,1]$!!!!
What am I doing wrong?
Thanks!
To make sense of the OP's question, let us agree to strike out the 2nd part of his question that begins with "Using Bayes:". Also, we feel it is necessary to frame the question in a way that makes more sense.
We are interested in obtaining statistical estimates for a discrete random variable $X$ with outcomes in $\{1,2,3\}$.
Statistician $A$ comes up with
$\tag A (\frac{1}{3},\frac{1}{3},\frac{1}{3})$
and statistician $B$ comes up with
$\tag B (\frac{2}{6},\frac{1}{6},\frac{3}{6})$.
We trust each of the statisticians the same amount, so we simply take the average, $C = \frac{A+B}{2}$, to estimate $X$:
$\tag C (\frac{4}{12},\frac{3}{12},\frac{5}{12})$
I am wondering where the OP found this question. But no matter, it is now necessary for us to commission another statistician, $C$, to put this matter to rest (or for the OP to work on clarifying the question).