Tools for Probability

Tree Diagrams

A tree diagram shows all the outcomes as branches. It is particularly helpful if events take on several stages where the occurrence of certain events depends on which events occurred previously.

Suppose A and B are two dependent events, where Event B depends on whether Event A occurs or not. The following tree diagram shows all the possible outcomes:

Example

The faculty has 60% tenured professors and 40% junior professors.

The probability that a professor wins a teaching award, given that they are a tenured professor, is 37%:

The probability that a professor wins a teaching award, given that they are a junior professor, is 45%:

We can organize this information in a tree diagram:

Contingency Table (Two-way Table)

A contingency table or two-way table is great for helping you with questions that calculating probabilities involving two events A and B. The row totals and column totals always add up to 1:

Notes

• $P\left(A\right)+P\left(A^c\right)=1$
• $P\left(B\right)+P\left(B^c\right)=1$
• $P\left(A\cap B\right)+P\left(A\cap B^c\right)+P\left(A^c\cap B\right)+P\left(A^c\cap B^c\right)=1$

Example

Event A and Event B are two events where $P(A)=0.7$ and $P(B)=0.5$.

$P(A\cap B)$ must be between ________ and ________, inclusive.