No answers to refer to, so would be really greatful if someone can take a look at this problem and explain me a bit where am I going wrong or if I've done something right to begin with, haha.
A fair coin is tossed tree times. Specific the probability space for the experiment and consider the following events: A: The result of the third toss is heads B: the coin shows heads exactly two times
Compute the conditional probabilities P(A given B) and P(B given A). Are A and B independent?
So our sample space is: ${HHH}, {TTT}, {HHT}, {HTT}, {THH}, {TTH}, {HTH}, {THT}$
Hence, $P(A): 0.5$
$P(B): 0.375$
We can see that event A appears exactly 2 times when event B happens, hence, A is independent of B as the conditional probability of A given B is still equal to the probability of event A= 0.5
However, I'm a bit confused for the second part in finding the conditional probability of B given A. B is appearing also twice while A is taking place, but that is different than the probability of event B alone, so is it true that B is dependent on A? 0.66 versus the P(B) = 0.375. If so, how do we compute the conditional probability? Any help would be much much aprpeciated!