Write sample spaces for the experiments in Exercises 3

Write sample spaces for the experiments in Exercises 3–10.

A box contains five balls, numbered 1, 2, 3, 4, and 5. A ball is drawn at random, the number on it recorded, and the ball replaced. The box is shaken, a second ball is drawn, and its number is recorded.

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Key Concepts

Replacement

Replacement is a procedure in probability experiments where an item is returned to the original set before the next trial. This process ensures that the conditions of the experiment, such as the composition of the sample space, remain unchanged throughout all trials.

Sample Space

The sample space in probability theory is the set of all possible outcomes of an experiment. It is a fundamental concept that provides a complete list of outcomes, ensuring that every potential result is considered when calculating probabilities or analyzing experimental data.

Independent Trials

Independent trials refer to successive experiments where the outcome of one does not influence the outcome of the other. In contexts like drawing with replacement, the result of the first draw does not affect the subsequent draws, making each trial independent and preserving the consistency of the sample space.

Cartesian Product

The Cartesian product is a mathematical operation that returns all ordered pairs (or tuples) from two or more sets. In the context of repeated experiments with replacement, the sample space of the experiment is the Cartesian product of the individual sets of outcomes, thereby representing every possible combination of outcomes across the trials.

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