Question

But change it to be a biased coin where Pr(flipping tails) = 0.25 and Pr(flipping heads) = 0.75

Answer 1

In case of a biased coin, P(H)=0.75 P(T)=0.25

When Head comes up a single die is rolled, giving us only one chance in 6 of getting 3

So P(3|H)=1/6

When tail comes up, we roll 2 dice and we get three as a sum of two dice.

2+1 or 1+2. So we have 2/36 chance of getting a total of 3 in a roll of two dice.

That is P(3|T)=2/36

Now we want to get

$P(H|3)=\frac{P(3|H)*P(H)}{P(3|H)*P(H)+P(3|T)*P(T)}$

$P(H|3)=\frac{1/6*0.75}{1/6*0.75+2/36*0.25}=0.9$