Tree Diagrams in Probability
Senior 2 · Mathematics
Tree Diagrams in Probability
Lesson 2 of 4
20 min
At a school canteen you choose a starch (rice or potatoes) and then a sauce (beans, peas or isombe). How many different meals are possible? When an experiment happens in stages, listing outcomes in your head is easy to get wrong. A tree diagram lays out every path, one stage after another, so that nothing is missed and nothing is counted twice.
Start from a point on the left. Draw one branch for each possible result of the first stage. From the end of each branch, draw one branch for each possible result of the second stage, and so on. Each complete path from left to right is one outcome; write it at the end of the path.
Worked example 1
A coin is tossed twice.
Figure 1. Tree diagram for tossing a coin twice. Diagram: Ganzaa.
There are four paths, so the sample space is {HH, HT, TH, TT}. Notice that HT (head, then tail) and TH (tail, then head) are different outcomes.
If the first stage has m results and the second stage has n results for each of them, the tree has m × n paths. With more stages, keep multiplying.
When all the paths are equally likely, count the paths in the event and divide by the total number of paths.
Worked example 2
A coin is tossed twice. Find P(exactly one head).
The paths HT and TH have exactly one head: 2 of the 4 paths, so P = 24 = 12.
Worked example 3
A fair coin is tossed and a fair die is thrown. Find P(a head and an even number).
12 equally likely outcomes; the event is {H2, H4, H6}, 3 outcomes. P = 312 = 14.
Tip
One stage per column, every path to the end. Label the top of each column ("1st toss", "2nd toss") and write each outcome at the end of its path. Order matters: HT and TH are two different paths.
A coin is tossed three times. How many outcomes are there? List them.
Solution: 2 × 2 × 2 = 8: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.
In question 1, find P(at least two heads).
Solution: HHH, HHT, HTH, THH: 4 of 8, so P = 48 = 12.
In question 1, find P(three heads).
Solution: only HHH: 18.
Spinner A shows 1, 2 or 3 and spinner B shows 1 or 2, each equally likely. Both are spun. Find P(the total is 4).
Solution: 3 × 2 = 6 outcomes. A total of 4 comes from (2, 2) and (3, 1): P = 26 = 13.
Remember
Sources and credits
The Rwandan examples, practice items and exercises are written by Ganzaa.
