Tree diagrams and contingency tables

Each branch shows an outcome and its probability.

• Multiply along a path for "and".
• Add the results of different paths for "or".
• The probabilities on branches from one point add up to 1.

Without replacement the second-stage probabilities change, because there is one fewer item.

Example. A bag has 5 red and 3 blue counters. Two are drawn without replacement.

$P(\text{both red})=\frac{5}{8}\times \frac{4}{7}=\frac{5}{14}$

$P(\text{one of each})=\frac{5}{8}\times \frac{3}{7}+\frac{3}{8}\times \frac{5}{7}=\frac{15}{28}$

A contingency table shows counts for two categories, with row and column totals.

Example. 200 learners were asked whether they prefer online or in-person tutoring.

• Boys: online 48, in-person 32, total 80
• Girls: online 72, in-person 48, total 120
• Totals: online 120, in-person 80, grand total 200

Read probabilities straight from the table:

• $P(\text{girl and in-person})=\frac{48}{200}=0,24$
• $P(\text{online})=\frac{120}{200}=0,6$

Complete any missing cells first using the totals.

Two events are independent if $P(A\text{ and }B)=P(A)\times P(B)$.

From a table. In the example above:

$P(\text{boy})\times P(\text{online})=\frac{80}{200}\times \frac{120}{200}=0,24$

$P(\text{boy and online})=\frac{48}{200}=0,24$

They are equal, so gender and preference are independent in this sample.

From a tree. If the second-stage branch probabilities are the same whatever happened first, the events are independent. If they change, the events are dependent.

Always show both calculations and write a conclusion in words.