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?

1. 0.445 or 0.5

2. 0.167 or 0.2

3. Yes and No

Step-by-step explanation:

2 coins are fair and the 3rd is two-headed

So there are 2 tails and 4 heads altogether.

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

The probability that the coin is a fair one is: Probability of a fair coin x Probability of obtaining a head

= 2/3 x 4/6 = 0.445 to 3 decimal places

(B) He randomly picks one and flips it twice. Compute the probability that he gets 2 tails

This is = [the sum of the probability that each coin produces a tail 2 times when flipped twice] divided by 3.

For Coin 1,  0.5 is the probability of getting tails. For two consecutive tails, the probability would be 0.5 x 0.5 = 0.25

Same goes for Coin 2 which is the second fair coin.

For Coin 3, the probability of getting a tail at all is 0.

So [0.25 + 0.25 + 0] / 3  = 0.167 to 3 d.p.

(C) Suppose B = the event that the first result is head

Suppose C = the event that the second result is also head

- Are B and C independent (if the two-headed coin wasn't picked)?

YES

- Are B and C independent (if the two-headed coin was picked)?

NO