- Describe and find conditional probabilities.
- Understand the concept of independent events.
Independent and Mutually Exclusive Events
- For two events to be independent, the outcome of one event does not impact the outcome of a successive event. Tossing a fair coin or rolling a fair die are often considered independent events. Just because you rolled a [latex]1[/latex] does not change the probability the next roll will be a [latex]1[/latex].
- Sampling with replacement is associated with independent events.
- Sampling without replacement is associated with dependent events.
- If two events are mutually exclusive, that means they cannot happen at the same time with a single outcome. Two events are mutually exclusive if the probability of both events happening at the same time is zero. For example, consider flipping a coin. It can land heads up or heads down, but it cannot be both heads up and heads down simultaneously. Thus, heads and tails are mutually-exclusive events.
Mutually exclusive events are NOT independent events.
Mutually exclusive events are events that cannot happen together, while independent events are events where the occurrence of one does not affect the other.
Event [latex]A[/latex] and event [latex]B[/latex] are independent and mutually exclusive events if only if [latex]P(A) = 0[/latex] and/or [latex]P(B) = 0[/latex]. This is because mutually exclusive means [latex]P(A \text{ and } B)=0[/latex] and independent events means [latex]P(A \text{ and } B) = P(A) \times P(B)[/latex].
Therefore, [latex]0 = P(A) \times P(B)[/latex] and this is only true if [latex]P(A) = 0[/latex] and/or [latex]P(B) = 0[/latex].