So the problem goes something like this:
There are 3 coins in a bag (C1, C2, C3). C1 comes up heads $1/3$ of the time, C2 $1/2$ of the time, and C3 $3/4$ of the time. Now you choose a coin at random, flip it, record the outcome and without replacing the coin back in the bag, pick another coin (out of the 2 remaining) and flip it.
a) What is the probability that 1 heads and 1 tails is flipped and b) what is the probability that a heads was flipped on the 1st flip given a tails was flipped on the 2nd.
I have figured out how to go about this if we put the coin back into the bag. In that case the 1 heads and 1 tails is $P(H_1T_2) + P(T_1H_2)$ which using a probability tree I found to be 95/216 (not sure if this is correct, just what I got as an answer). Same way for b) i found the answer to be 228/432 when you put the coin back into the bag.
I am not sure how to go about finding the solution to when the coin is not put back into the bag. Any help would be appreciated.
I am fairly new to probability so if the problem isn't well stated I apologize.