You are rolling a 6-sided die, but you don't know whether it's fair or not. You know that that are only two kinds of dice: fair ones and ones that land on 5 40% of the time, and 1,2,3,4,6 each 12% of the time. Let's call these Hypothesis 1 and Hypothesis 2.
H1: Die is fair
H2: Die lands on 5 40% of the time, and 1,2,3,4,6 each 12% of the time
Your prior for H1 is 0.5 and your prior for H2 is 0.5.
a) Given just these priors, what is the probability that you will roll a 5?
You want to determine which kind of die you have, so you rol it 4 times, and get the following: 5,2,5,5,
b) What is the probability of this sequence of rolls if H1 is true? What if H2 is true?