Probability of Independent and Mutually Exclusive Events

Understanding Probability

Probability is a measure of how likely an event is to occur. It ranges from 0 (impossible event) to 1 (certain event).

Independent Events

• Definition: Two events are independent if the occurrence of one does not affect the occurrence of the other.
• Formula: For two independent events, A and B, the probability of both occurring is given by:
  $$P(A \text{ and } B) = P(A) \times P(B)$$

Example

If you flip a coin and roll a die, the probability of getting a head and a 4 is:

• $$P(\text{Head}) = \frac{1}{2}$$
• $$P(4) = \frac{1}{6}$$
• $$P(\text{Head and 4}) = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12}$$

Mutually Exclusive Events

• Definition: Two events are mutually exclusive if they cannot occur at the same time.
• Formula: For two mutually exclusive events, A and B, the probability of either occurring is given by:
  $$P(A \text{ or } B) = P(A) + P(B)$$

Example

If you roll a die, the probability of getting a 2 or a 5 is:

• $$P(2) = \frac{1}{6}$$
• $$P(5) = \frac{1}{6}$$
• $$P(2 \text{ or } 5) = \frac{1}{6} + \frac{1}{6} = \frac{1}{3}$$