Let’s Summarize

To summarize the relationship between two categorical variables, use:

• Data display: contingency table, tree diagram or Venn diagram
• Numerical summaries: probabilities

Keys Ideas from Our Work with Probability

• The probability of an event is a measure of the likelihood that the event occurs. Probabilities are always between [latex]0[/latex] and [latex]1[/latex]. The closer the probability is to [latex]0[/latex], the less likely the event is to occur. The closer the probability is to [latex]1[/latex], the more likely the event is to occur.
• A sample space or contingency tables can be used to calculate basic probabilities, where you count the number of items of interest and divide by the total number of items.
• [latex]P(A \ \mathrm{and} \ B)[/latex] means events [latex]A[/latex] and [latex]B[/latex] must happen in the same outcome. If [latex]A[/latex] and [latex]B[/latex] are independent events, then [latex]P(A \ \mathrm{and} \ B) = P(A)P(B)[/latex].
• [latex]P(A \ \mathrm{or} \ B)[/latex] means either event [latex]A[/latex] or [latex]B[/latex] (or both) must happen in the outcome.
  • If [latex]A[/latex] and [latex]B[/latex] are any two mutually exclusive events, then [latex]P(A \ \mathrm{OR} \ B) = P(A) + P(B)[/latex].
  • If [latex]A[/latex] and [latex]B[/latex] are NOT mutually exclusive events, then [latex]P(A \ \mathrm{OR} \ B) = P(A) + P(B) - P(A \ \mathrm{and} \ B)[/latex].
• A conditional probability is calculated based on the assumption that one event has already occurred. A conditional probability for event [latex]A[/latex] given event [latex]B[/latex] has happened is calculated as: [latex]P(A|B) = \frac{P(A \ \mathrm{and} \ B)}{P(B)}[/latex]