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Assessment · Probability Trees
20 marks
Answer all questions. Show all your working; you get marks for method. Work on your own.Name: __________ Class: ______ Date: ______
A bag contains 2 green and 3 yellow balls. A ball is taken at random, its colour noted and replaced, then a second is taken. The tree diagram is partly complete. Find the probability marked $?$ (the second pick being yellow after a green).
Answer: (1 mark)
A box contains 3 black and 7 white beads. A bead is taken at random, its colour noted and replaced, then a second is taken. The tree diagram is partly complete. Find the probability marked $?$ (the second pick being white after a black).
Answer: (1 mark)
A bag contains 8 red and 2 blue counters. A counter is taken at random, its colour noted and replaced, then a second is taken. The tree diagram is partly complete. Find the probability marked $?$ (the second pick being blue after a red).
Answer: (1 mark)
A jar contains 6 toffee and 2 mint sweets. A sweet is taken at random, its colour noted and replaced, then a second is taken. The tree diagram is partly complete. Find the probability marked $?$ (the second pick being mint after a toffee).
Answer: (1 mark)
A bag contains 4 green and 6 yellow balls. A ball is taken at random, its colour noted and replaced, then a second is taken. The tree diagram is partly complete. Find the probability marked $?$ (the second pick being yellow after a green).
Answer: (1 mark)
A jar contains 2 toffee and 3 mint sweets. A sweet is taken at random, its colour noted and replaced, then a second is taken. The tree diagram is partly complete. Find the probability marked $?$ (the second pick being mint after a toffee).
Answer: (1 mark)
A bag contains 2 red and 3 blue counters. A counter is taken at random, its colour noted and replaced, then a second is taken. The tree diagram is partly complete. Find the probability marked $?$ (the second pick being blue after a red).
Answer: (1 mark)
A box contains 2 black and 6 white beads. A bead 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 at least one is black.
Answer: (3 marks)
A jar contains 3 toffee and 2 mint sweets. A sweet is taken at random, its colour noted and replaced, then a second is taken. Complete the tree diagram. Use it to find the probability that the first is toffee and the second is mint.
Answer: (3 marks)
The probability a basketball player scores a free throw is 0.2, independently each attempt. Two are considered. Complete the tree diagram. Use it to find the probability that the event happens on both days/attempts.
Answer: (3 marks)
A spinner has outcomes 1, 2 and 3 with probabilities $\dfrac{1}{8}$, $\dfrac{3}{8}$ and $\dfrac{1}{2}$. It is spun twice. Complete the first stage of the tree diagram. Use it to find the probability of getting 2 both times.
Answer: (2 marks)
A bag contains 2 green and 4 yellow balls. A ball 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 at least one is green.
Answer: (3 marks)
A box contains 2 black and 4 white beads. A bead is taken at random, its colour noted and replaced, then a second is taken. Complete the tree diagram. Use it to find the probability that the first is black and the second is white.
Answer: (3 marks)
The probability a train is on time on any day is 0.2, independently each day. Two are considered. Complete the tree diagram. Use it to find the probability that the event happens on exactly one of the two.
Answer: (3 marks)
A spinner has outcomes red, green and blue with probabilities $\dfrac{1}{5}$, $\dfrac{3}{10}$ and $\dfrac{1}{2}$. It is spun twice. Complete the first stage of the tree diagram. Use it to find the probability of getting green both times.
Answer: (2 marks)
A bag contains 2 green and 4 yellow balls. A ball 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 green.
Answer: (3 marks)
A box contains 3 black and 2 white beads. A bead is taken at random, its colour noted and replaced, then a second is taken. Complete the tree diagram. Use it to find the probability that the first is black and the second is white.
Answer: (3 marks)
A jar contains 2 toffee and 3 mint sweets. A sweet is taken at random, its colour noted and replaced, then a second is taken. Complete the tree diagram. Use it to find the probability that the two sweets are different colours.
Answer: (3 marks)
Bag A has 2 toffee and 5 mint sweets. Bag B has 4 toffee and 2 mint sweets. One sweet is taken at random from each bag. Complete the tree diagram. Use it to find the probability that one sweet is toffee and the other is mint.
Answer: (3 marks)
A jar contains 3 toffee and 2 mint sweets. A sweet is taken at random, its colour is noted and it is put back. A second sweet is then taken. The tree diagram is partly complete. Write down the probability marked $?$ on the tree diagram.
Answer: (1 mark)
Spinner A lands on red with probability 0.25. Spinner B lands on red with probability 0.65. Each spinner is spun once. Write down the probability marked $?$ on the tree diagram.
Answer: (1 mark)
A jar contains 2 toffee and 3 mint sweets. A sweet is taken at random, its colour noted and it is put back; a second is then taken. Complete the tree diagram. Work out the probability that both sweets are toffee.
Answer: (2 marks)
A bag contains 7 red and 3 blue counters. Two counters are picked at random. Complete the tree diagram. Use it to find the probability that both are red.
Answer: (3 marks)
A part is made on machine X with probability $\dfrac{3}{5}$, otherwise on machine Y. A part from machine X is faulty with probability $\dfrac{4}{5}$. A part from machine Y is faulty with probability $\dfrac{1}{5}$. Complete the tree diagram. Use it to find the probability that a part is from machine X and is faulty.
Answer: (3 marks)
A bag contains 5 red and 3 blue counters. Two counters are picked at random. Complete the tree diagram. Work out the probability that both are red.
Answer: (2 marks)
A bag contains 3 red and 4 blue counters. Two counters are picked at random. Complete the tree diagram. Use it to find the probability that at least one counter is red.
Answer: (4 marks)
Bag A contains 4 red counters out of 6. Bag B contains $n$ of its 5 counters coloured red. One counter is taken at random from each bag. The probability that both are red is $\dfrac{8}{15}$. Work out the value of $n$.
Answer: (3 marks)
The probability the traffic light is red is 0.8. If it is red, the probability the train is late is 0.7. If it is not red, the probability the train is late is 0.4. Write down the probability marked $?$ on the tree diagram.
Answer: (1 mark)
Mark scheme · Probability Trees
Total: 20 marks
Award the marks shown for each correct step, then add up the total out of 20. A method mark counts even if the final answer is wrong.
A bag contains 2 green and 3 yellow balls. A ball is taken at random, its colour noted and replaced, then a second is taken. The tree diagram is partly complete. Find the probability marked $?$ (the second pick being yellow after a green).[1 mark]
A box contains 3 black and 7 white beads. A bead is taken at random, its colour noted and replaced, then a second is taken. The tree diagram is partly complete. Find the probability marked $?$ (the second pick being white after a black).[1 mark]
A bag contains 8 red and 2 blue counters. A counter is taken at random, its colour noted and replaced, then a second is taken. The tree diagram is partly complete. Find the probability marked $?$ (the second pick being blue after a red).[1 mark]
A jar contains 6 toffee and 2 mint sweets. A sweet is taken at random, its colour noted and replaced, then a second is taken. The tree diagram is partly complete. Find the probability marked $?$ (the second pick being mint after a toffee).[1 mark]
A bag contains 4 green and 6 yellow balls. A ball is taken at random, its colour noted and replaced, then a second is taken. The tree diagram is partly complete. Find the probability marked $?$ (the second pick being yellow after a green).[1 mark]
A jar contains 2 toffee and 3 mint sweets. A sweet is taken at random, its colour noted and replaced, then a second is taken. The tree diagram is partly complete. Find the probability marked $?$ (the second pick being mint after a toffee).[1 mark]
A bag contains 2 red and 3 blue counters. A counter is taken at random, its colour noted and replaced, then a second is taken. The tree diagram is partly complete. Find the probability marked $?$ (the second pick being blue after a red).[1 mark]
A box contains 2 black and 6 white beads. A bead 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 at least one is black.[3 marks]
P(at least one black) $= 1 - \dfrac{9}{16} = \dfrac{7}{16}$.
Answer: $\dfrac{7}{16}$
Marks
✔1 markP(neither) = $\dfrac{9}{16}$
✔1 mark1 − P(neither)
✔1 mark$\dfrac{7}{16}$
A jar contains 3 toffee and 2 mint sweets. A sweet is taken at random, its colour noted and replaced, then a second is taken. Complete the tree diagram. Use it to find the probability that the first is toffee and the second is mint.[3 marks]
Method
Independent picks (replaced), so P(toffee) = $\dfrac{3}{5}$ and P(mint) = $\dfrac{2}{5}$ each time.
P(toffee then mint) $= \dfrac{3}{5} \times \dfrac{2}{5} = \dfrac{6}{25}$.
Answer: $\dfrac{6}{25}$
Marks
✔1 markRead the toffee-then-mint route
✔1 mark$\dfrac{3}{5} \times \dfrac{2}{5}$
✔1 mark$\dfrac{6}{25}$
The probability a basketball player scores a free throw is 0.2, independently each attempt. Two are considered. Complete the tree diagram. Use it to find the probability that the event happens on both days/attempts.[3 marks]
Method
The events are independent, so multiply along the branches.
P(both scores) $= 0.2 \times 0.2 = 0.04$.
Answer: $0.04$
Marks
✔1 markIdentify independent multiply
✔1 mark0.2 × 0.2
✔1 mark0.04
A spinner has outcomes 1, 2 and 3 with probabilities $\dfrac{1}{8}$, $\dfrac{3}{8}$ and $\dfrac{1}{2}$. It is spun twice. Complete the first stage of the tree diagram. Use it to find the probability of getting 2 both times.[2 marks]
A bag contains 2 green and 4 yellow balls. A ball 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 at least one is green.[3 marks]
P(at least one green) $= 1 - \dfrac{4}{9} = \dfrac{5}{9}$.
Answer: $\dfrac{5}{9}$
Marks
✔1 markP(neither) = $\dfrac{4}{9}$
✔1 mark1 − P(neither)
✔1 mark$\dfrac{5}{9}$
A box contains 2 black and 4 white beads. A bead is taken at random, its colour noted and replaced, then a second is taken. Complete the tree diagram. Use it to find the probability that the first is black and the second is white.[3 marks]
Method
Independent picks (replaced), so P(black) = $\dfrac{1}{3}$ and P(white) = $\dfrac{2}{3}$ each time.
P(black then white) $= \dfrac{1}{3} \times \dfrac{2}{3} = \dfrac{2}{9}$.
Answer: $\dfrac{2}{9}$
Marks
✔1 markRead the black-then-white route
✔1 mark$\dfrac{1}{3} \times \dfrac{2}{3}$
✔1 mark$\dfrac{2}{9}$
The probability a train is on time on any day is 0.2, independently each day. Two are considered. Complete the tree diagram. Use it to find the probability that the event happens on exactly one of the two.[3 marks]
Method
P(on time then late) $= 0.2 \times 0.8 = 0.16$; same for the other order.
P(exactly one on time) $= 2 \times 0.16 = 0.32$.
Answer: $0.32$
Marks
✔1 markTwo mixed routes
✔1 mark2 × 0.16
✔1 mark0.32
A spinner has outcomes red, green and blue with probabilities $\dfrac{1}{5}$, $\dfrac{3}{10}$ and $\dfrac{1}{2}$. It is spun twice. Complete the first stage of the tree diagram. Use it to find the probability of getting green both times.[2 marks]
Method
P(green) $= \dfrac{3}{10}$ each spin (independent).
✔1 markP(green) = $\dfrac{3}{10}$ read from the tree
✔1 mark$\dfrac{9}{100}$
A bag contains 2 green and 4 yellow balls. A ball 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 green.[3 marks]
Method
The events are independent (replaced), so each pick has P(green) = $\dfrac{1}{3}$.
A box contains 3 black and 2 white beads. A bead is taken at random, its colour noted and replaced, then a second is taken. Complete the tree diagram. Use it to find the probability that the first is black and the second is white.[3 marks]
Method
Independent picks (replaced), so P(black) = $\dfrac{3}{5}$ and P(white) = $\dfrac{2}{5}$ each time.
P(black then white) $= \dfrac{3}{5} \times \dfrac{2}{5} = \dfrac{6}{25}$.
Answer: $\dfrac{6}{25}$
Marks
✔1 markRead the black-then-white route
✔1 mark$\dfrac{3}{5} \times \dfrac{2}{5}$
✔1 mark$\dfrac{6}{25}$
A jar contains 2 toffee and 3 mint sweets. A sweet is taken at random, its colour noted and replaced, then a second is taken. Complete the tree diagram. Use it to find the probability that the two sweets are different colours.[3 marks]
Method
P(toffee then mint) $= \dfrac{6}{25}$; P(mint then toffee) $= \dfrac{6}{25}$.
Bag A has 2 toffee and 5 mint sweets. Bag B has 4 toffee and 2 mint sweets. One sweet is taken at random from each bag. Complete the tree diagram. Use it to find the probability that one sweet is toffee and the other is mint.[3 marks]
Method
P(toffee from 1st, mint from 2nd) $= \dfrac{2}{7} \times \dfrac{1}{3} = \dfrac{2}{21}$.
P(mint from 1st, toffee from 2nd) $= \dfrac{5}{7} \times \dfrac{2}{3} = \dfrac{10}{21}$.
P(one of each) $= \dfrac{2}{21} + \dfrac{10}{21} = \dfrac{4}{7}$.
Answer: $\dfrac{4}{7}$
Marks
✔1 markTwo mixed routes
✔1 markAdd the routes
✔1 mark$\dfrac{4}{7}$
A jar contains 3 toffee and 2 mint sweets. A sweet is taken at random, its colour is noted and it is put back. A second sweet is then taken. The tree diagram is partly complete. Write down the probability marked $?$ on the tree diagram.[1 mark]
Spinner A lands on red with probability 0.25. Spinner B lands on red with probability 0.65. Each spinner is spun once. Write down the probability marked $?$ on the tree diagram.[1 mark]
Method
The branches from each point add to 1.
Missing probability $= 1 - 0.65 = 0.35$.
Answer: $0.35$
Marks
✔1 mark1 − 0.65 = 0.35
A jar contains 2 toffee and 3 mint sweets. A sweet is taken at random, its colour noted and it is put back; a second is then taken. Complete the tree diagram. Work out the probability that both sweets are toffee.[2 marks]
Method
Fill every branch: each pick has P(toffee) $= \dfrac{2}{5}$ (replaced, so independent); multiply along the top route.
A bag contains 7 red and 3 blue counters. Two counters are picked at random. Complete the tree diagram. Use it to find the probability that both are red.[3 marks]
Method
Two picked together (nothing put back) means WITHOUT replacement – the second pick has one fewer counter, denominator 9.
First pick: P(red) $= \dfrac{7}{10}$. After removing one red, 6 red remain of 9.
A part is made on machine X with probability $\dfrac{3}{5}$, otherwise on machine Y. A part from machine X is faulty with probability $\dfrac{4}{5}$. A part from machine Y is faulty with probability $\dfrac{1}{5}$. Complete the tree diagram. Use it to find the probability that a part is from machine X and is faulty.[3 marks]
Method
Tree: P(machine X) $= \dfrac{3}{5}$, P(machine Y) $= \dfrac{2}{5}$; then P(faulty) $= \dfrac{4}{5}$ after machine X but $\dfrac{1}{5}$ after machine Y.
Multiply along the machine X-then-faulty branch: $\dfrac{3}{5} \times \dfrac{4}{5} = \dfrac{12}{25}$.
Answer: $\dfrac{12}{25}$
Marks
✔1 markCorrect probabilities on the tree (both conditionals)
✔1 mark$\dfrac{3}{5} \times \dfrac{4}{5}$
✔1 mark$\dfrac{12}{25}$
A bag contains 5 red and 3 blue counters. Two counters are picked at random. Complete the tree diagram. Work out the probability that both are red.[2 marks]
Method
Fill every branch: without replacement the second-pick probabilities change to a denominator of 7.
✔1 markSecond pick is $\dfrac{4}{7}$ (one fewer of 7)
✔1 mark$\dfrac{5}{14}$
A bag contains 3 red and 4 blue counters. Two counters are picked at random. Complete the tree diagram. Use it to find the probability that at least one counter is red.[4 marks]
Method
Two picked together (nothing put back) means WITHOUT replacement – the second pick has one fewer counter, denominator 6.
P(at least one red) $= 1 - \dfrac{2}{7} = \dfrac{5}{7}$.
Answer: $\dfrac{5}{7}$
Marks
✔1 markP(both blue) = $\dfrac{2}{7}$
✔1 markRecognise 1 − P(none)
✔1 markSubtract from 1
✔1 mark$\dfrac{5}{7}$
Bag A contains 4 red counters out of 6. Bag B contains $n$ of its 5 counters coloured red. One counter is taken at random from each bag. The probability that both are red is $\dfrac{8}{15}$. Work out the value of $n$.[3 marks]
So $\dfrac{4n}{30} = \dfrac{8}{15}$, giving $4n = 16$.
Therefore $n = 4$.
Answer: $4$
Marks
✔1 markSet up $\dfrac{2}{3} \times \dfrac{n}{5} = \dfrac{8}{15}$
✔1 markForm the equation $4n = 16$
✔1 mark$n = 4$
The probability the traffic light is red is 0.8. If it is red, the probability the train is late is 0.7. If it is not red, the probability the train is late is 0.4. Write down the probability marked $?$ on the tree diagram.[1 mark]
Method
Given it is not red, P(late) = 0.4 is stated in the question.
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