Practice Tree diagrams and multi-stage random experiments
A tree diagram organises the outcomes of a multistep experiment along branches.
Multiply probabilities along a path and add them across mutually exclusive paths.
Exercises with solutions
A tree diagram represents multi-stage random experiments: from each node the possible outcomes of the next stage branch off. The probability of a path is found by multiplying the probabilities along that path. If several paths belong to one event, you add their probabilities.
Exercise 1
A coin is tossed twice. How many paths does the tree diagram have?
Show solution and working
4
Each stage has two outcomes, so 2 · 2 = 4 paths: HH, HT, TH and TT.
Exercise 2
What is the probability of heads twice?
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1/4
Multiply along the path: 1/2 · 1/2 = 1/4.
Exercise 3
What is the probability of exactly one heads?
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1/2
The favourable paths are HT and TH, each with probability 1/4. So 1/4 + 1/4 = 1/2.
Exercise 4
When do you add path probabilities?
Show solution and working
when several paths belong to the same event
Within one path you multiply; several paths of the same event are added.
Common mistake
Swapping the two rules. Within a path you multiply, across several paths you add. And not checking which paths actually belong to the event: “exactly one heads” excludes HH and TT.
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