23.4 Combined events and tree diagrams — Learn

Prerequisites: fractions, multiplication and possibility spaces.

Objectives: use a tree diagram for two events; multiply probabilities along a path; add probabilities for separate successful paths.

A tree diagram shows each possible result in sequence. For independent events, the first result does not change the probabilities on the next set of branches. Multiply along a path to find the probability of that outcome. Add the probabilities of paths that meet the event.

Example 1. A fair coin is tossed twice. Find the probability of at least one head.

The tree has four equally likely paths: HH, HT, TH and TT.

Three paths contain at least one head: HH, HT and TH.

The complement gives the same result: $1-P(\text{no heads})=1-\frac14=\frac34$.

Example 2. A bag contains 3 red and 2 blue counters. One counter is drawn and not replaced. Find the probability of red then blue.

The probability of red first is $\frac35$. After a red is removed, 2 blue counters remain among 4, so the probability of blue next is $\frac24$.

The second probability changes because the first counter is not replaced. These are dependent events: the first draw changes the probability of the second.

Watch out. For “at least one”, do not count only HH. Include every path with one or more heads.

Progress check

The spinner is spun twice. Draw a tree diagram and work out the probability of red both times. Then work out the probability of at least one blue. Explain how the complement gives the second answer.

You should now be able to

• draw branches for sequential events;
• identify whether events are independent or dependent;
• multiply along a path and add successful paths to find combined probabilities.