Consider the following marble game: A box contains m red marbles and n blue marbles (n, m > 0) and a fixed number - here three - of draws. The rules of the draw are as follows:
(i) If a red marble is drawn, it is placed in a temporary holding pan which is outside the box.
(ii) If a blue marble is drawn, it is placed back in the box along with all of the (necessarily red) marbles in the holding pan.
Aside from the above, the rules/law is according to the usual random selections described in similar contexts.
Let us denote by b1 the event of a blue marble on the first draw and similarly for b2, . . . , r3. Show that the 1 and 3 type events are not independent but, conditioned on b2, these events are (conditionally) independent.