Summary: Probability

Key Equations

probability of an event with equally likely outcomes [latex]P\left(E\right)=\dfrac{n\left(E\right)}{n\left(S\right)}[/latex]
probability of the union of two events [latex]P\left(E\cup F\right)=P\left(E\right)+P\left(F\right)-P\left(E\cap F\right)[/latex]
probability of the union of mutually exclusive events [latex]P\left(E\cup F\right)=P\left(E\right)+P\left(F\right)[/latex]
probability of the complement of an event [latex]P\left(E\text{'}\right)=1-P\left(E\right)[/latex]

Key Concepts

  • Probability is always a number between 0 and 1, where 0 means an event is impossible and 1 means an event is certain.
  • The probabilities in a probability model must sum to 1.
  • When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in the sample space for the experiment.
  • To find the probability of the union of two events, we add the probabilities of the two events and subtract the probability that both events occur simultaneously.
  • To find the probability of the union of two mutually exclusive events, we add the probabilities of each of the events.
  • The probability of the complement of an event is the difference between 1 and the probability that the event occurs.
  • In some probability problems, we need to use permutations and combinations to find the number of elements in events and sample spaces.

Glossary

complement of an event the set of outcomes in the sample space that are not in the event [latex]E[/latex]

event any subset of a sample space

experiment an activity with an observable result

mutually exclusive events events that have no outcomes in common

outcomes the possible results of an experiment

probability a number from 0 to 1 indicating the likelihood of an event

probability model a mathematical description of an experiment listing all possible outcomes and their associated probabilities

sample space the set of all possible outcomes of an experiment

union of two events the event that occurs if either or both events occur