Probability
Tree diagrams and combined events
In this note
- 8.3Draw and use tree diagrams for two (or three) events.
- 8.3Use the AND rule (multiply) and the OR rule (add).
- 8.3Handle picks with replacement and without replacement Extended.
Before you start: Sample spaces and Venn diagrams and multiplying fractions.
Formula bank
AND and OR
Two events are independent if one doesn't change the chance of the other (two coin flips; picks WITH replacement). If Sara is late with probability 0.2 on any day: late on both of two days \(=0.2\times0.2=0.04\). Late on exactly one day = late-then-on-time OR on-time-then-late \(=0.2\times0.8+0.8\times0.2=0.32\).
General methodHow to answer any tree-diagram question
- Draw the first set of branches with their probabilities (they add to 1).
- From each end, draw the second set. Without replacement: one fewer item in total, and one fewer of the colour just taken.
- Multiply ALONG each route that gives what you want.
- Add the routes you want. 'At least one'? Use 1 − P(none).
Without replacement Extended
A bag has 4 red and 6 blue counters. Two are taken without replacement, so the second pick has only 9 counters left, and one fewer of the colour that was taken. (With replacement the second branches would be the same as the first: 4/10 and 6/10.)
Example Find the probability that both counters are red.
Route R then R: multiply along it.
- \(\tfrac{4}{10}\times\tfrac39=\tfrac{12}{90}\)
Answer \(\tfrac{2}{15}\)
Example Find the probability that the counters are different colours.
Two routes: RB or BR. Multiply along each, then add.
- \(\tfrac{4}{10}\times\tfrac69+\tfrac{6}{10}\times\tfrac49=\tfrac{24}{90}+\tfrac{24}{90}\)
Answer \(\tfrac{48}{90}=\tfrac{8}{15}\)
Example Find the probability that at least one counter is red.
1 − P(no red) = 1 − P(BB).
- \(P(BB)=\tfrac{6}{10}\times\tfrac59=\tfrac13\)
- \(1-\tfrac13\)
Answer \(\tfrac23\)
Try it yourself
Show answer
Answer 0.16
Show answer
Answer 0.48
Ways they test it
Type 1 Complete the tree
"Complete the tree diagram." (2 marks)
Spot it a tree with gaps.
- Each pair of branches adds to 1.
- Without replacement: change the second branches.
Type 2 Both / neither
"Find the probability that both are red." (2 marks)
Spot it 'both', 'neither'.
- One route; multiply.
For example 4 R, 6 B, no replacement
- \(\tfrac4{10}\times\tfrac39\)
Answer \(\tfrac{2}{15}\)
Type 3 One of each / exactly one
"Find the probability that they are different colours." (3 marks)
Spot it 'one of each', 'exactly one'.
- Two routes.
- Multiply each, add.
Type 4 At least one
"Find the probability that at least one is red." (3 marks)
Spot it 'at least one'.
- 1 − P(none).
Type 5 Without a tree
"Three dice are rolled. Find P(three sixes)." (2 marks)
Spot it three or more events.
- Multiply the probabilities in a line.
Mistakes to avoid
- ✗ Adding along a branch✓ multiply ALONG, add BETWEEN routes
- ✗ Same second branches without replacement✓ change both the total and the colour taken
- ✗ Only one route for 'one of each'✓ RB AND BRBoth orders count.
- ✗ Long route list for 'at least one'✓ 1 − P(none)Faster and fewer slips.
Summary
- Multiply along branches (and); add between routes (or).
- Without replacement: second total is one smaller, and so is the colour taken.
- At least one = 1 − P(none).
If you see …
- 'without replacement'change the second branches.
- 'one of each'two routes.
Practice
Practice: easy, medium, hard
Easy
Show answer
Answer 0.04
Show answer
Answer 1/216
Show answer
Answer 0.36
Medium
Guide me
What is it asking? a probability
What do we know? LO or OL
Plan \(2\times0.2\times0.8\)
Show working
- 0.32
Answer 0.32
Guide me
What is it asking? a probability
What do we know? 1 − P(never late)
Plan \(1-0.8^{2}\)
Show working
- 0.36
Answer 0.36
Guide me
What is it asking? a probability
What do we know? 6/10 then 5/9
Plan multiply
Show working
- \(\tfrac{30}{90}\)
Answer 1/3
Hard
Guide me
What is it asking? a probability
What do we know? 5/8 then 4/7
Plan multiply
Show working
- \(\tfrac{20}{56}\)
Answer 5/14
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(\tfrac58\times\tfrac47\) | M1 | ||
| \(\tfrac{5}{14}\) | A1 | ||
| Total: 2 marks | |||
Guide me
What is it asking? a probability
What do we know? RG + GR
Plan \(2\times\tfrac38\times\tfrac57\)
Show working
- \(\tfrac{30}{56}\)
Answer 15/28
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(\tfrac38\times\tfrac57\) | M1 | ||
| × 2 or both routes | M1 | ||
| \(\tfrac{15}{28}\) | A1 | ||
| Total: 3 marks | |||
Guide me
What is it asking? a probability
What do we know? 1 − P(BB)
Plan \(1-\tfrac13\)
Show working
- \(\tfrac23\)
Answer 2/3
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(1-\tfrac6{10}\times\tfrac59\) | M1 | ||
| \(\tfrac23\) | A1 | ||
| Total: 2 marks | |||
Can I do this?
- I can multiply along branches and add routes.
- I can draw a tree with and without replacement.
- I can use 1 − P(none) for 'at least one'.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
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