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 $0$ and $1$. The closer the probability is to $0$, the less likely the event is to occur. The closer the probability is to $1$, 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.
• $P(A \ \mathrm{and} \ B)$ means events $A$ and $B$ must happen in the same outcome. If $A$ and $B$ are independent events, then $P(A \ \mathrm{and} \ B) = P(A)P(B)$.
• $P(A \ \mathrm{or} \ B)$ means either event $A$ or $B$ (or both) must happen in the outcome.
  • If $A$ and $B$ are any two mutually exclusive events, then $P(A \ \mathrm{OR} \ B) = P(A) + P(B)$.
  • If $A$ and $B$ are NOT mutually exclusive events, then $P(A \ \mathrm{OR} \ B) = P(A) + P(B) - P(A \ \mathrm{and} \ B)$.
• A conditional probability is calculated based on the assumption that one event has already occurred. A conditional probability for event $A$ given event $B$ has happened is calculated as: $P(A|B) = \frac{P(A \ \mathrm{and} \ B)}{P(B)}$