Conditional probabilityAQA A-Level Maths: Subtopic test

1. 1
   In a group of 120 students, 70 study Biology, 50 study Chemistry and 20 study both. Twenty students study neither subject. One student is chosen at random.

   (a)

   Given that the student studies Biology, find the probability that they also study Chemistry.

   [1 mark]

   • A$\frac{2}{7}$
   • B$\frac{2}{5}$
   • C$\frac{1}{6}$
   • D$\frac{5}{12}$

   (b)

   Given that the student does not study Chemistry, find the probability that they study Biology.

   [1 mark]

   • A$\frac{7}{12}$
   • B$\frac{5}{12}$
   • C$\frac{2}{7}$
   • D$\frac{5}{7}$

   (c)

   Determine whether studying Biology and studying Chemistry are independent.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A bag contains 5 red counters and 3 blue counters. Two counters are taken at random, one after the other, without replacement.

   (a)

   Given that the first counter is blue, find the probability that the second counter is also blue.

   [1 mark]

   • A$\frac{3}{8}$
   • B$\frac{3}{7}$
   • C$\frac{2}{7}$
   • D$\frac{1}{4}$

   (b)

   Find the probability that the two counters are of different colours.

   [1 mark]

   • A$\frac{15}{56}$
   • B$\frac{15}{28}$
   • C$\frac{3}{7}$
   • D$\frac{5}{14}$

   (c)

   Given that the second counter is red, find the probability that the first counter was blue.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   In a survey of 80 people, 45 own a bicycle, 30 own a car and 12 own both.

   (a)

   Find the probability that a person chosen at random owns neither a bicycle nor a car. Given that they own at least one of the two, find the probability that they own a bicycle.

   [3 marks]

   (b)

   Two different people are chosen at random without replacement. Find the probability that the first owns a bicycle and the second owns a car.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In a school, 70% of students revise for a test. A student who revises passes with probability $0.9$ and a student who does not revise passes with probability $0.4$. Students are independent of one another.

   (a)

   A student is chosen at random. Find
   (i) the probability that the student passes,
   (ii) the probability that the student revised, given that they passed,
   (iii) the probability that the student did not revise, given that they failed.

   [6 marks]

   (b)

   Three students are chosen at random. Find the probability that exactly two pass, and the probability that exactly two pass given that at least one passes.

   [6 marks]

   Total for question 4: 12 marks

End of questions