Basic Concepts of Probability

Probability is the measure of the likelihood that an event will occur. It quantifies uncertainty and is always between 0 and 1.

• $P(E) = 0$ → Event is impossible
• $P(E) = 1$ → Event is certain

1. Experiment

An experiment is a process or action that leads to one or more outcomes.

Examples:

• Tossing a coin
• Rolling a die
• Drawing a card from a deck

2. Sample Space (S)

The sample space is the set of all possible outcomes of an experiment.

Examples:

• Tossing a coin: $S = \{H, T\}$
• Rolling a die: $S = \{1, 2, 3, 4, 5, 6\}$

3. Event (E)

An event is a subset of the sample space. It represents one or more outcomes.

Examples:

• Rolling an even number: $E = \{2,4,6\}$
• Tossing a head: $E = \{H\}$

4. Types of Events

5. Probability Rules

1. Classical (Theoretical) Probability $$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$$

• Example: Probability of rolling 3 on a die = 1/6

2. Complementary Rule $$P(E') = 1 - P(E)$$

• $E'$ = event does not occur

3. Addition Rule (for mutually exclusive events)

   $$P(A \cup B) = P(A) + P(B)$$

4. General Addition Rule

   $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

5. Multiplication Rule (for independent events)

   $$P(A \cap B) = P(A) \cdot P(B)$$

6. Important Properties

• $0 \leq P(E) \leq 1$
• $P(S) = 1$
• $P(\emptyset) = 0$
• $P(A) + P(A') = 1$

7. Types of Probability

Key Points to Remember

• Probability is always between 0 and 1.
• Classical probability = favorable outcomes ÷ total outcomes.
• Complement, addition, and multiplication rules are essential for solving problems.