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 $A$ and $B$ are independent if one of the following is true:

1. $P(A|B) = P(A)$
2. $P(B|A) = P(B)$
3. $P(A \ \mathrm{AND} \ B) = P(A)P(B)$

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

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