In probability, independent and dependent events are two categories of events that describe how the occurrence of one event affects the occurrence of another. Here's a breakdown of the key differences:
Independent Events
- Definition: Two events are considered independent if the occurrence of one event does not affect the occurrence of the other.
- Probability Calculations: For independent events A and B, the probability of both events occurring is given by the product of their probabilities: \[ P(A \text{ and } B) = P(A) \times P(B) \]
- Example: Flipping a coin and rolling a die are independent events. The outcome of the coin flip (heads or tails) has no influence on the outcome of the die roll (any number from 1 to 6).
Dependent Events
- Definition: Two events are considered dependent if the occurrence of one event affects the occurrence of the other. In other words, the outcome of one event changes the probability of the other event.
- Probability Calculations: For dependent events A and B, the probability of both events occurring is given by: \[ P(A \text{ and } B) = P(A) \times P(B | A) \] Here, \(P(B | A)\) is the conditional probability of B occurring given that A has occurred.
- Example: Drawing cards from a deck without replacement is an example of dependent events. If you draw an Ace from a standard deck of 52 cards, the probability of drawing another Ace on the next draw changes (it becomes 3 out of 51 instead of 4 out of 52).
Summary
- Independent Events: Their outcomes do not affect each other, and the joint probability can be calculated using the product of their individual probabilities.
- Dependent Events: Their outcomes are related, and the probability must account for how the occurrence of one event affects the probability of the other.