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.

(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.