Suppose you have a bag that contains three six-sided dice; two of the dice are fair, but the third die is irregular, having probabilities given in the following table:
Face 1 2 3 4 5 6
P(Face) 0.15 0.15 0.15 0.15 0.25 0.15
That is, the outcome "5" has higher probability with the irregular die compared to a fair die. Suppose further that these three dice look and feel identical. Randomly choose a die from the bag, roll this die 10 times, and let X be the number of 5s observed.
(a) What is the probability of seeing four 5s? [Hints: (i) The type of die chosen partitions the sample space; (ii) if you know what type of die is being rolled, the number of 5s is a Binomial RV.]
(b) If X = 4 was observed, what is the probability the irregular die was rolled?