Probability & Statistics – Problem 2: Determine which statements are correct (multiple choice): A

A bag contains 4 balls of identical size and texture: 2 white and 2 black. Two draws are made in sequence with replacement.

Event A (Jia): the first draw is white.

Event B (Yi): the second draw is black.

Event C (Bing): both draws are white.

Determine which statements are correct (multiple choice):

A. Events A and B are mutually exclusive.

B. Events B and C are mutually exclusive.

C. Events A and B are independent.

D. Events A and B are complementary.

With replacement, the equally likely outcomes are $$\{WW,WB,BW,BB\}.$$

1) A and B can occur together (outcome $WB$), so they are not mutually exclusive. Therefore A is false.

2) B means the second draw is black, while C means both draws are white; they cannot occur simultaneously. Therefore B is true.

3) Let A be "first white" and B be "second black": $$P(A)=\frac12,\quad P(B)=\frac12,\quad P(AB)=P(WB)=\frac14.$$ Since $$P(AB)=P(A)P(B),$$ A and B are independent, so C is true.

4) Complementary events must be mutually exclusive and exhaustive. A and B are not mutually exclusive, so D is false.

Hence the correct options are $B,C$.

Because the sampling is with replacement, the four outcomes $WW,WB,BW,BB$ are equally likely. Events A and B are not mutually exclusive (they overlap at $WB$), but they are independent since $P(AB)=\frac14=P(A)P(B)$.

Event B and event C are mutually exclusive, while A and B are not complementary because they are neither disjoint nor exhaustive as a pair.

Therefore, the correct option set is $B,C$.