Q. 1
You have two bags. One bag contains 2 red balls and 2 white balls. The other contains 1 red ball and 7 white balls. We will call them ‘the 2-2 bag’ and ‘the 1-7 bag,’ respectively. You select a bag uniformly at random. You then make two draws (taking a ball out) from that bag with replacement, so that it is possible for you to draw the same ball twice in a row.
Let B denote the event that you have chosen the 2-2 bag, so its complement Bc is the event that you have chose the 1-7 bag. Let Rj denote the event that your jth draw is red, for j = 1 or 2.
a) Find P(B | R1).
b) Are the events R1 and R2 independent? Prove your answer.
Hint: One method is to find P(R1 \cap R2 | B), and likewise given Bc, then use these to find P(R1 \cap R2).