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Probability

Represent chance numerically, use equally likely outcomes and combine simple events.

12 free questions · Notes · Worked solutions

Practise this topic

What you will practise

  • Express probabilities from 0 to 1 and as fractions or percentages.
  • Calculate probabilities for equally likely outcomes.
  • Use complementary events and expected frequency.
  • Combine independent events using multiplication and tree diagrams.

Revision notes

Probability scale

Probability ranges from 0 (impossible) to 1 (certain). For equally likely outcomes, P(event)=number of favourable outcomes/number of possible outcomes.

Complements

The probability an event does not happen is 1−P(event). Probabilities of all mutually exclusive outcomes in a complete sample space add to 1.

Combined events

For independent events, multiply probabilities along a tree branch. For mutually exclusive alternatives, add branch probabilities. A tree diagram labels each branch with conditional probability at that stage.

Expected frequency

Expected frequency=number of trials×probability. It is a long-run average prediction, not a guarantee for a small number of trials.

A worked example

A bag contains 5 red and 3 blue counters. One counter is selected and replaced, then a second is selected. Find the probability both are red.

  1. There are 8 counters, so P(red on the first selection)=5/8.
  2. The counter is replaced, so P(red on the second selection)=5/8 as well.
  3. The selections are independent: (5/8)(5/8)=25/64.

25/64

Try it yourself

These are original practice questions, not official past-paper questions. Practice points count correct final answers; working is not automatically assessed.

Choose a question to begin.
Question 1 · Standard1 practice point

A fair six-sided die is rolled once. What is P(rolling an even number)?

Need a hint?

Three of the six faces are even.

Worked solution
  1. Even outcomes are 2, 4, 6: three outcomes.
  2. P=3/6=1/2.
Question 2 · Standard1 practice point

An event has probability 0.2. What is the probability it does not happen?

Need a hint?

Use 1−P(event).

Worked solution
  1. Complement probability=1−0.2.
  2. It equals 0.8.
Question 3 · Standard1 practice point

A bag has 2 green and 6 yellow counters. A counter is picked at random. What is P(green)?

Need a hint?

Compare green counters with the total.

Worked solution
  1. There are 8 counters in total.
  2. P(green)=2/8=1/4.
Question 4 · Standard1 practice point

Two independent events have probabilities 0.4 and 0.5. What is the probability both happen?

Need a hint?

Multiply probabilities for independent events both occurring.

Worked solution
  1. P(both)=0.4×0.5.
  2. P(both)=0.2.
Question 5 · Foundation1 practice point

A spinner has 8 equal sections, 3 of which are marked blue. Find P(blue).

Need a hint?

Use blue sections over all sections.

Worked solution
  1. P(blue)=3/8.
  2. 3÷8=0.375.
Question 6 · Foundation2 practice points

A fair coin is tossed 40 times. Find the expected number of heads.

Need a hint?

Expected frequency is trials multiplied by probability.

Worked solution
  1. P(heads)=1/2.
  2. Expected heads=40×1/2=20.
Question 7 · Foundation1 practice point

A box contains 4 white and 6 black balls. One ball is selected. Find the probability it is not black.

Need a hint?

Not black means white.

Worked solution
  1. There are 10 balls altogether.
  2. P(not black)=4/10=0.4.
Question 8 · Standard2 practice points

A fair six-sided die is rolled twice. Find the probability of getting a 6 both times. Give your answer as a decimal to 3 significant figures.

Need a hint?

The two rolls are independent.

Worked solution
  1. Each roll has P(6)=1/6.
  2. P(two sixes)=1/6×1/6=1/36.
  3. As a decimal to 3 significant figures, this is 0.0278.
Question 9 · Stretch3 practice points

A bag contains 3 red and 7 blue counters. One is selected and not replaced. Find the probability that the first is red and the second is blue. Give your answer as a decimal to 3 significant figures.

Need a hint?

After a red counter is removed, 9 counters remain.

Worked solution
  1. P(first red)=3/10.
  2. Then 7 blue of 9 remaining: P(second blue|first red)=7/9.
  3. Multiply: 3/10×7/9=7/30≈0.233 to 3 significant figures.
Question 10 · Standard2 practice points

The probability that a bus is late is 0.15. Over 200 journeys, find the expected number of late arrivals.

Need a hint?

Multiply 200 by 0.15.

Worked solution
  1. Expected number=200×0.15.
  2. =30 journeys.
Question 11 · Stretch3 practice points

A fair spinner has 5 equal sectors numbered 1 to 5. It is spun twice. Find the probability that the two numbers have a sum of 7.

Need a hint?

List ordered pairs that add to 7; there are 25 equally likely pairs.

Worked solution
  1. Favourable ordered pairs are (2,5), (3,4), (4,3), (5,2).
  2. There are 4 favourable pairs out of 5×5=25.
  3. P=4/25=0.16.
Question 12 · Standard2 practice points

Events A and B are mutually exclusive, with P(A)=0.35 and P(B)=0.2. Find P(A or B).

Need a hint?

Add probabilities of mutually exclusive events.

Worked solution
  1. They cannot occur together, so P(A or B)=P(A)+P(B).
  2. 0.35+0.20=0.55.

Coverage and sources

Selected IGCSE concepts, with a mix of introductory and more demanding questions. Check your own specification. AI drafted; independent teacher review is pending. Editorial details.

Cambridge IGCSE Mathematics 0580 syllabus

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