Question

Franklin has three coins, two fair coins (head on one side and tail on the other side) and one two-headed coin.

1) He randomly picks one, flips it and gets a head. What is the probability that the coin is a fair one?

2)  He randomly picks one, flips it twice. Compute the probability that he gets two tails.

3) He randomly picks one and flips it twice. Suppose B stands for the event that the first result is head, and C represents the event that the second result is also head. Are B and C independent? Are B and C independent, conditioned on the event that the two-headed coin was picked?

Answer 1

Define the events

The following probabilities are known.

.

1) Using total probability theorem,

The conditional probability,

2) The following probabilities are known.

.

Using total probability theorem,

3) Define the events

The event .

The event .  

Using total probability theorem,

Since , the events A, C are not independent.

When the two-headed coin was picked,

since .

Hence A and C independent, conditioned on the event that the two-headed coin was picked.

Note: Int this part, I have used A to represent event B, since B is already used.