A box contains 4 balls numbered 1 to 4. Record a sample space using a tree diagram for the following experiments:

(A) A ball is drawn, and the number is recorded. Then the ball is returned, and a second ball is drawn and recorded.

(B) A ball is drawn and recorded. Without replacing the first ball, the experimenter draws and records a second ball.

(C) What are the sizes of these two sample spaces?

(A) Tree diagram:

$$S = \{(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4), (4, 1), (4, 2), (4, 3), (4, 4)\}.$$

$$\text{Size} = 4 \times 4 = 16.$$

(B) Tree diagram:

First draw has 4 branches; second draw has only 3 (the same ball cannot appear twice).

$$S = \{(1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (2, 4), (3, 1), (3, 2), (3, 4), (4, 1), (4, 2), (4, 3)\}.$$

$$\text{Size} = 4 \times 3 = 12.$$

(C) With replacement: $n(S) = 16$ .

$$\text{Without replacement: } n(S) = 12.$$