A box contains 4 balls numbered 1 to 4. Record a sample space using a tree diagram for the following experiments:
A ball is drawn, and the number is recorded. Then the ball is returned, and a second ball is drawn and recorded.
A tree diagram visually represents all possible outcomes of a multi-step experiment, where each branch represents a possible outcome and each path from start to end represents one complete outcome. For the experiment with replacement, the tree diagram starts with 4 branches for the first draw (1, 2, 3, 4). Since the ball is returned, each first branch splits into 4 branches for the second draw (1, 2, 3, 4). The sample space recorded is S₁ = {(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)}.\n\n
A ball is drawn and recorded. Without replacing the first ball, the experimenter draws and records a second ball.
For the experiment without replacement, the tree diagram again starts with 4 branches for the first draw (1, 2, 3, 4). However, since the first ball is not replaced, each first branch splits into only 3 branches for the second draw, excluding the number already drawn. For example, if the first draw is 1, the second draw branches are 2, 3, and 4. The sample space recorded is S₂ = {(1,2), (1,3), (1,4), (2,1), (2,3), (2,4), (3,1), (3,2), (3,4), (4,1), (4,2), (4,3)}.\n\n
What are the sizes of these two sample spaces?
The size of a sample space is the total number of possible outcomes. For the first experiment (with replacement), there are 4 outcomes for the first draw and 4 for the second, giving a size of = 16. For the second experiment (without replacement), there are 4 outcomes for the first draw and 3 remaining outcomes for the second, giving a size of = 12.
Explanation
The question requires constructing sample spaces using tree diagrams for two different multi-step experiments: drawing with replacement and drawing without replacement. In part (i), because the ball is returned, the second draw is independent of the first, resulting in 4 x 4 = 16 outcomes. In part (ii), because the ball is not replaced, the second draw depends on the first, resulting in 4 x 3 = 12 outcomes. Part (iii) simply asks for the sizes of these respective sample spaces, which are 16 and 12.