Revision mat · Probability Trees

No calculator13 marks

Question mix weighted by 153 real exam questions from 92 GCSE papers (2017–24), so the most common question types get the most space. Show your working.

Independent≈65% of real exam Qs
1A jar contains 8 toffee and 2 mint sweets.
A sweet is taken at random, its colour noted, and put back.
Then a second is taken.
Complete the tree diagram.
Use it to find the probability that both are toffee.
toffeeminttoffeeminttoffeemint
Answer: (3 marks)
Without replacement≈20% of real exam Qs
2A box contains 6 black and 3 white beads.
Two beads are taken at random without replacement.
Complete the tree diagram.
Use it to find the probability that both are black.
blackwhiteblackwhiteblackwhite
Answer: (3 marks)
3A jar contains 3 toffee and 3 mint sweets.
Two sweets are taken at random without replacement.
Complete the tree diagram.
Use it to find the probability that one is toffee and one is mint.
toffeeminttoffeeminttoffeemint
Answer: (3 marks)
★ Exam capstonemixed & other forms ≈15% of real exam Qs
4A bag contains $n$ red counters and 2 blue counters.
Two counters are taken at random without replacement.
The probability that both counters are red is $\dfrac{3}{10}$.
Work out the value of $n$.
Answer: (4 marks)

Mark scheme · Probability Trees

Total: 13 marks

Award the marks shown for each correct step, then add up the total out of 13. A method mark counts even if the final answer is wrong.

1A jar contains 8 toffee and 2 mint sweets.
A sweet is taken at random, its colour noted, and put back.
Then a second is taken.
Complete the tree diagram.
Use it to find the probability that both are toffee.[3 marks]
Method
The events are independent (replaced), so each pick has P(toffee) = $\dfrac{4}{5}$.
P(both toffee) $= \dfrac{4}{5} \times \dfrac{4}{5} = \dfrac{16}{25}$.
Answer: $\dfrac{16}{25}$
Marks
✔1 markP(toffee) = $\dfrac{4}{5}$ both picks
✔1 markMultiply along the branches
✔1 mark$\dfrac{16}{25}$
2A box contains 6 black and 3 white beads.
Two beads are taken at random without replacement.
Complete the tree diagram.
Use it to find the probability that both are black.[3 marks]
Method
First pick: P(black) $= \dfrac{2}{3}$. After removing one black, 5 black remain of 8.
P(both black) $= \dfrac{2}{3} \times \dfrac{5}{8} = \dfrac{5}{12}$.
Answer: $\dfrac{5}{12}$
Marks
✔1 markSecond-pick probabilities change to /8
✔1 mark$\dfrac{2}{3} \times \dfrac{5}{8}$
✔1 mark$\dfrac{5}{12}$
3A jar contains 3 toffee and 3 mint sweets.
Two sweets are taken at random without replacement.
Complete the tree diagram.
Use it to find the probability that one is toffee and one is mint.[3 marks]
Method
P(toffee then mint) $= \dfrac{1}{2} \times \dfrac{3}{5}$; P(mint then toffee) $= \dfrac{1}{2} \times \dfrac{3}{5}$.
Add the two routes: P(one of each) $= \dfrac{3}{5}$.
Answer: $\dfrac{3}{5}$
Marks
✔1 markTwo routes, denominators 6 then 5
✔1 markAdd both routes
✔1 mark$\dfrac{3}{5}$
4A bag contains $n$ red counters and 2 blue counters.
Two counters are taken at random without replacement.
The probability that both counters are red is $\dfrac{3}{10}$.
Work out the value of $n$.[4 marks]
Method
Let there be $n$ red counters and 2 blue, so $5$ in total.
P(both red) $= \dfrac{n}{5} \times \dfrac{n-1}{4} = \dfrac{n(n-1)}{20}$.
Set $\dfrac{n(n-1)}{20} = \dfrac{3}{10}$, so $n(n-1) = 6$.
Solving gives $n = 3$ (since $3 \times 2 = 6$).
Answer: $3$
Marks
✔1 mark$\dfrac{n}{5} \times \dfrac{n-1}{4}$
✔1 markForms $n(n-1) = 6$
✔1 markSolves the quadratic
✔1 mark$n = 3$
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