Conditional probability

Probability and statistics

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 $7$ blue counters. Find the probability that a counter taken at random is red.

2 marks
The first counter is red and is not replaced. Find the probability that the second counter is also red.

2 marks
Five red cards and three blue cards are in a pile. Two cards are taken at random without replacement. Find the probability that both are red.

3 marks
A class has $12$ girls and $8$ boys. Five girls and two boys wear glasses. Find the probability that a student wears glasses given that the student is a girl, and the probability that a student with glasses is a girl.

3 marks
The probability of rain is $0.3$. The probability of being late is $0.4$ if it rains and $0.1$ if it does not. Find the probability of rain and being late, and the probability of being late.

4 marks
A bag contains $4$ red beads and $6$ yellow beads. Two beads are taken without replacement. Find the probability that they are the same colour.

4 marks
Of $30$ students, $18$ study French. One third of the French students also study Spanish, and half of the other students study Spanish. Find the probability that a student studies Spanish, and the probability that a Spanish student also studies French.

5 marks
A box has $5$ milk chocolates and $3$ dark chocolates. Two are eaten at random without replacement. Find the probability that both are milk, the probability of one of each, and the probability that the second is milk given that the first was dark.

6 marks

Mark scheme

Answer. $\dfrac{3}{10}$
- There are $10$ counters.
- $\dfrac{3}{10}$
Answer. $\dfrac{2}{9}$
- Two red counters remain, out of $9$.
- $\dfrac{2}{9}$
Answer. $\dfrac{5}{14}$
- $\dfrac{5}{8} \times \dfrac{4}{7} = \dfrac{20}{56}$
- $\dfrac{5}{14}$
Answer. $\dfrac{5}{12}$ and $\dfrac{5}{7}$
- Of the girls, $\dfrac{5}{12}$ wear glasses.
- Seven students wear glasses, of whom $5$ are girls, so $\dfrac{5}{7}$.
Answer. $0.12$ and $0.19$
- $0.3 \times 0.4 = 0.12$
- $0.7 \times 0.1 = 0.07$
- Total $= 0.19$
Answer. $\dfrac{7}{15}$
- $\dfrac{4}{10} \times \dfrac{3}{9} + \dfrac{6}{10} \times \dfrac{5}{9} = \dfrac{42}{90}$
- $\dfrac{7}{15}$
Answer. $\dfrac{2}{5}$ and $\dfrac{1}{2}$
- Spanish students $= 18 \times \dfrac{1}{3} + 12 \times \dfrac{1}{2} = 12$.
- $\dfrac{12}{30} = \dfrac{2}{5}$
- Of those $12$, $6$ study French, so the conditional probability is $\dfrac{1}{2}$.
Answer. $\dfrac{5}{14}$, $\dfrac{15}{28}$ and $\dfrac{5}{7}$
- Both milk $= \dfrac{5}{8} \times \dfrac{4}{7} = \dfrac{5}{14}$.
- One of each $= 2 \times \dfrac{5}{8} \times \dfrac{3}{7} = \dfrac{15}{28}$.
- After a dark chocolate, $5$ of the remaining $7$ are milk, so $\dfrac{5}{7}$.