Question

1. A fair coin is tossed twice. Consider the following events:
a) A = a head on the first toss;
b) B = a tail on the first toss;
c) C = a tail on the second toss;
d) D = a head on the second toss.

Are events A and B mutually exclusive ? WHY or WHY NOT ?

Are events C and D Independent ? WHY or WHY NOT ?

2. Suppose there are 3 similar boxes. Box i contains i white ball(s) and two black balls. I pick one box at random, then pick a ball at random from the box. If the ball drawn is black, what is the probability that the ball came from Box i ?

( so, Box 1 contains 1 white ball and two black balls; Box 2 contains 2 white balls and two black
balls; Box 3 contains 3 white balls and two black balls.)

Answer 1

1.

If two events are mutually exclusive, it means that they cannot occur at the same time

therefore we can say A and B are mutually exclusive if there is heads in first toss (A) then tails (B) cannot be there in first toss

just like A and B , C and are also mutually exclusive

and,

mutually exclusive events are always dependent. The definition of independence for events A and B is that P(A and B) = P(A)P(B). However, in the case that A and B are mutually exclusive, then P(A and B) = 0.

so as we know C and D are mutually exclusive therefore C and D are not independent

2.

P(box i picked) = 1/3 {all 3 boxes have equal chance of being picked}

P(box i picked and black ball picked) = P(box i) * P(black ball | box i) = (1/3)*(no. of black ball / total balls) = (1/3)*(2 / (2+i))

P(black) = sum of ( P(box i picked and black ball picked) )

= (1/3)*(2 / (2+1)) + (1/3)*(2 / (2+2)) + (1/3)*(2 / (2+3))

= 0.5222

P(box i | black) = P(black ball | box i)*P(box i) / P(black)

= (2 / (2+i))*(1/3) / 0.5222

= 2 / ((0.5222*3)*(2+i))

P(box i | black) = 1.2766 / (2+i)

P.S. (please upvote if you find the answer satisfactory)