Two counters drawn without replacement from one of two boxes chosen by tossing coins: distributions, a tree diagram, a total probability and a reverse conditional probability

Box \(P\) holds \(3\) blue counters and \(4\) white counters. Two counters are taken from box \(P\) at random, without replacement. Let \(X\) be the number of blue counters taken. The probability distribution of \(X\) is given in the table.

1. Calculate \(\mathrm{E}(X)\), the expected number of blue counters taken.[3]

Box \(Q\) holds \(5\) blue counters and \(3\) white counters. Two counters are taken from box \(Q\) at random, without replacement. Let \(Y\) be the number of blue counters taken.

2. 1. Draw a tree diagram to show the possible colours of the two counters taken from box \(Q\), with the probability on each branch.
   2. Hence find the probability distribution of \(Y\).[8]

A fair coin is tossed twice. If it shows heads both times, two counters are taken from box \(P\). Otherwise, two counters are taken from box \(Q\).

3. Find the probability that both counters taken are blue.[5]
4. Given that both counters taken are blue, find the probability that the coin showed heads both times.[3]