Q.A bag contains four cards marked as 5, 6, 7 and 8. Find the sample space if two cards are drawn at random a) with replacement b) without replacement.

Random Experiment and Sample Space

Start with intuition

Think about flipping a coin. Before you flip it, you know the only possible outcomes are heads or tails. But you cannot say which one will actually come up — that depends on chance. That's the core idea: a random experiment is any process where you know what could happen, but you don't know exactly what will happen in a single trial.

Another example: rolling a die. You know the result will be one of 1, 2, 3, 4, 5, or 6. But before you roll, you cannot predict the exact number. That's randomness.

The sample space is simply the set of all possible outcomes of that random experiment. For the coin, the sample space is $\{H, T\}$. For the die, it's $\{1, 2, 3, 4, 5, 6\}$.

The precise definition

A random experiment satisfies three conditions:

1. It can be repeated any number of times under essentially the same conditions.
2. All possible outcomes are known in advance.
3. The actual outcome on any given trial cannot be predicted with certainty.

The sample space (denoted by $S$ or $\Omega$) is the set of all possible outcomes of a random experiment. Each individual outcome is called a sample point.

Examples to cement the idea