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Probability

Probability questions range from a simple 'chance of picking a red counter' to hard-tier problems with combined events and no replacement. Answers are usually fractions in their simplest form, so accurate fraction work matters as much as the idea.

What is tested

  • Probability as a fraction from a given set of outcomes
  • Probabilities that add to 1 (the complement)
  • Combined events: for two independent events (one does not affect the other), P(A and B) = P(A) × P(B); if the first changes the second (taking without replacement), update the numbers before multiplying. If two events cannot happen together, P(A or B) = P(A) + P(B); if they can overlap, subtract the overlap: P(A) + P(B) − P(A and B)
  • Two picks with and without replacement

Worked example

A bag holds 4 red and 6 blue counters. Two are taken without replacement. What is the probability that both are red?

  1. First pick red: 4 out of 10 = 4/10.
  2. Now 3 red are left out of 9 counters: 3/9.
  3. Both red: 4/10 × 3/9 = 12/90.
  4. Simplify: 12/90 = 2/15.

Answer: 2/15

Exam-day checklist

  • Write the total number of outcomes first — and update it if nothing is put back.
  • 'And' for independent events → multiply (update the numbers if nothing is put back); 'or' for events that cannot both happen → add. If they can overlap, the calculation is different.
  • Use 'not' = 1 − probability when it is easier.
  • Simplify the final fraction.

Common mistakes

  • Forgetting the total changes when an item is not replaced.
  • Adding probabilities when the question needs them multiplied.
  • Leaving the answer unsimplified.

Try it yourself

1. A spinner lands on red with probability 1/5 and blue with probability 3/10. Every other outcome is green. What is the probability of green?

Show answer

1/2

1/5 = 2/10, so red + blue = 2/10 + 3/10 = 5/10. Green = 1 − 5/10 = 5/10 = 1/2.

2. A fair die is rolled and a fair coin is flipped. What is the probability of an even number AND heads?

Show answer

1/4

P(even) = 3/6 = 1/2 and P(heads) = 1/2, so both = 1/2 × 1/2 = 1/4.

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