Assume that a gambler has two dice, one of which is fair, and the other is biased toward landing on $six$, so that $0.25$ of the time it lands on $six$, and $0.15$ of the time it lands on each of $1$, $2$, $3$, $4$ and $5$.
Question: The gambler chooses dice at random, and rolls it $six$ times, getting the values, $4$, $3$, $6$, $6$, $5$, $5$. The outcomes of rolls are mutually independent. What is the probability that fair die was chosen?
MY WORKING
Let $F$ denote the event that the fair die is chosen.
Let $F'$ denote the event that the biased die is chosen.
Let $X$ denote the event that die comes up with 4,3,6,6,5,5
We have to calculate $P(F|X)=?$
By Applying Bayes' Theorem we have:
$P(F|X)=\frac{P(X|F)P(F)}{P(X|F)P(F)+P(X|F')P(F')}$
Here we find that:
$P(F)=\frac{1}{2}$, $P(F')=\frac{1}{2}$
By conditional probability we have:
$P(X|F)=\frac{P(X\cap F)}{P(F)}$
$P(X|F')=\frac{P(X\cap F')}{P(F')}$
Here I am stuck here again. How do I solve the numerator part of above last two equations? Is there a relation between $X$, $F$ AND $X$, $F'$? I know that each roll in the event $X$ is independent, but what about the $X$, $F$ AND $X$, $F'$? I will appreciate any guidance here. Thanks
$$p(A|B) = \frac{p(A,B)}{p(B)}.$$
$A$ is the event that the die was fair.
$B$ is the event that the $6$ die rolls turned out as specified.
$$p(B) = p(A,B) + p(A^C,B).$$
$$p(A|B) = \frac{(1/2) \times (1/6)^6}{[(1/2) \times (1/6)^6] + [(1/2) \times (0.15)^4 \times (0.25)^2]}. \tag1 $$
In (1) above, to compute the probability of event $(A,B)$ occurring, first you have to compute the probability that the fair die was selected. Then, you have to compute the probability that the exact $6$ rolls occurred, given that the fair die was selected.
In other words, $p(A,B) = p(A) \times p(B|A).$
In (1) above, the analysis, re computing the probability of the event $(A^C,B)$ is virtually identical.
Edit
For what it's worth, I consider intuition to be a vital part of understanding Probability Theory. My intuition guided me through the convoluted analysis above. Intuition is developed by attacking many, many problems in the specific area of Math.