Random Experiment and Sample Space with Suitable Examples

Random Experiment

A random experiment is a process or action whose outcome cannot be predicted with certainty, even when performed under identical conditions. Every performance of such an experiment is called a trial. For example, tossing a fair coin or rolling a standard six-faced die are classic random experiments.

n(S) = total number of outcomes

Example: Solved Example: Identify the sample space of tossing two coins

Show Step-by-Step Solution

Step 1: List outcomes for coin 1: {H, T}.
Step 2: List outcomes for coin 2: {H, T}.
Step 3: Combine to get S = {HH, HT, TH, TT}.
Answer: n(S) = 4.

Sample Space

The sample space, denoted by S, is the set of all possible outcomes of a random experiment. Each element of the sample space is called a sample point. The number of elements in the sample space is denoted by n(S).

S = {ω₁, ω₂, ω₃, ..., ωₙ}

Example: Solved Example: Find the sample space of rolling a die and tossing a coin

Show Step-by-Step Solution

Step 1: Die outcomes D = {1, 2, 3, 4, 5, 6}.
Step 2: Coin outcomes C = {H, T}.
Step 3: S = {(1,H), (1,T), (2,H), (2,T), (3,H), (3,T), (4,H), (4,T), (5,H), (5,T), (6,H), (6,T)}.
Answer: n(S) = 12.

Events and Subsets

An event is a subset of the sample space S. If an experiment results in an outcome that is an element of the event E, we say the event has occurred. The empty set ∅ is an impossible event, and the set S itself is the sure event.

E ⊆ S

Example: Solved Example: Find event E of getting an even number on a die

Show Step-by-Step Solution

Step 1: S = {1, 2, 3, 4, 5, 6}.
Step 2: Identify even numbers in S: {2, 4, 6}.
Step 3: E = {2, 4, 6}.
Answer: n(E) = 3.