Tree diagrams
Generate theoretical sample spaces for single and combined events using listing, tables, and tree diagrams, and calculate theoretical probabilities as the number of favourable outcomes divided by the total number of equally likely outcomes
What a learner can do afterwards
- List all 36 outcomes when rolling two dice and find P(total = 7)
- Draw a tree diagram for two successive events and multiply along branches for combined probabilities
- Calculate the probability of a combined event using a sample space diagram and simplify the fraction
1 · Read
When a game starts with a coin flip and then a dice roll, it helps to see every possible turn before you start. A tree diagram is a map of every way things can turn out. Draw one branch for each outcome of the first event. Then grow new branches from the end of each one: one branch for each outcome of the second event, just as a table lists every pair for two dice. A coin has 2 outcomes and a die has 6, so each coin branch grows 6 die branches.
You do not always need the full drawing to count outcomes. Multiply the branch counts: 2 coin branches times 6 die branches gives 12 paths, and two dice give 6 times 6, 36 pairs. With three events, multiply all three counts. The complete set of outcomes is called the sample space. To find a probability, divide the favourable outcomes, the paths you want, by the total. If a question asks for red or blue, a union of two sets, count both kinds of path. An answer above 1 means the dividing step was missed.
Flip one coin, then flip it again. The first flip has 2 branches, Heads and Tails, and each one grows 2 more. That makes 4 paths: HH, HT, TH, TT, all equally likely. To find P(Heads then Tails), multiply along its path: 1/2 × 1/2 = 1/4. Counting agrees: HT is 1 path out of 4, and 1 out of 4 is 1/4.
In the 36-pair table for two dice, look for pairs with the same total. Try P(total of 10). The matching pairs are (4,6), (5,5), and (6,4): 3 favourable outcomes out of 36. Write 3/36, then simplify by dividing the top and bottom by 3, giving 1/12. Simplifying keeps the value the same; it only makes the fraction tidier.
Multiply along one path to get its probability, and add path probabilities when several paths count. As a final check, the probabilities of all the paths in a tree always add up to 1.
Sometimes the first item stays out, like a ball drawn and kept. Then the second fractions must be rebuilt from what is left. Take 3 red and 2 blue balls. After one red stays out, 2 red are left out of 4 balls, so the second draw uses 2/4, not 3/5. Multiply along the path as usual. You will see more of this later.
A tree diagram grows one branch per outcome at each step, so you can count every path. Multiply along a path for its probability, or divide favourable paths by the total, then simplify the fraction.
Adapted from OpenStax Statistics (CC-BY 4.0), openstax.org · license
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
16 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.