Tree diagrams

Probability and statistics

Questions marked Higher are aimed at the Higher tier.

These are original practice questions, not a past paper. This site is independent and not affiliated with AQA or Pearson.

A bag contains $3$ red counters and $2$ blue counters. A counter is taken and not replaced. Write the probabilities on the first branches.

Tree diagram for two counters taken without replacement

2 marks
Complete the second branches.

2 marks
Find the probability of two red counters.

3 marks
Find the probability of two blue counters.

3 marks
Find the probability of one counter of each colour.

4 marks
Find the probability that the second counter is red.

4 marks
The counters are now replaced after each pick. Find the probability of two red counters, one of each colour, and two blue counters. Higher

5 marks
The probability that a bus is late is $0.2$. If it is late, the probability of missing assembly is $0.7$. If it is on time, that probability is $0.1$. Find the probability of missing assembly, and the probability that the bus was late given that assembly was missed. Higher

6 marks

Mark scheme

Answer. Red $\dfrac{3}{5}$, blue $\dfrac{2}{5}$
- There are $5$ counters.
- $\dfrac{3}{5}$ and $\dfrac{2}{5}$
Answer. After red: $\dfrac{2}{4}$ and $\dfrac{2}{4}$. After blue: $\dfrac{3}{4}$ and $\dfrac{1}{4}$.
- After a red counter, $2$ red and $2$ blue remain.
- After a blue counter, $3$ red and $1$ blue remain.
Answer. $\dfrac{3}{10}$
- $\dfrac{3}{5} \times \dfrac{2}{4} = \dfrac{6}{20}$
- $\dfrac{3}{10}$
Answer. $\dfrac{1}{10}$
- $\dfrac{2}{5} \times \dfrac{1}{4} = \dfrac{2}{20}$
- $\dfrac{1}{10}$
Answer. $\dfrac{3}{5}$
- $\dfrac{3}{5} \times \dfrac{2}{4} + \dfrac{2}{5} \times \dfrac{3}{4} = \dfrac{12}{20}$
- $\dfrac{3}{5}$
Answer. $\dfrac{3}{5}$
- $\dfrac{3}{10} + \dfrac{3}{10}$
- $\dfrac{3}{5}$
Answer. $\dfrac{9}{25}$, $\dfrac{12}{25}$ and $\dfrac{4}{25}$
- Two red: $\dfrac{3}{5} \times \dfrac{3}{5} = \dfrac{9}{25}$.
- One of each: $2 \times \dfrac{3}{5} \times \dfrac{2}{5} = \dfrac{12}{25}$.
- Two blue: $\dfrac{4}{25}$.
Answer. $0.22$ and $\dfrac{7}{11}$
- Late and missed $= 0.2 \times 0.7 = 0.14$.
- On time and missed $= 0.8 \times 0.1 = 0.08$.
- Probability of missing assembly $= 0.22$.
- $\dfrac{0.14}{0.22} = \dfrac{7}{11}$