Key Concepts

• 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 1 does not change the probability the next roll will be a 1.
• 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.

Glossary

dependent events: if two events are NOT independent, then we say that they are dependent

independent events: the occurrence of one event has no effect on the probability of the occurrence of another event. Events [latex]A[/latex] and [latex]B[/latex] are independent if one of the following is true:

1. [latex]P(A|B) = P(A)[/latex]
2. [latex]P(B|A) = P(B)[/latex]
3. [latex]P(A \ \mathrm{AND} \ B) = P(A)P(B)[/latex]

mutually exclusive: two events are mutually exclusive if the probability that they both happen at the same time is zero. If events [latex]A[/latex] and [latex]B[/latex] are mutually exclusive, then [latex]P(A \ \mathrm{AND} \ B) = 0[/latex].

sampling with replacement: if each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once

sampling without replacement: when sampling is done without replacement, each member of a population may be chosen only once