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Probability Basics and Relative Frequency

Calculating simple probabilities, using the probability scale, and estimating from relative frequency and expected outcomes.

  • 9 key terms
  • All boards

Learning Objectives

  1. 1Use the probability scale from 0 to 1 and write probabilities as fractions, decimals or percentages.
  2. 2Calculate the probability of an event, and the probability that it does not happen.
  3. 3Use relative frequency to estimate probabilities, and work out expected numbers of outcomes.
  4. 4List outcomes systematically, and use the addition rule for mutually exclusive events.

Measuring chance

Probability tells you how likely something is, using a number from 0 to 1. Almost every probability question on a non-calculator paper uses neat numbers, such as a bag of 10 counters or two dice, so the answers come out as simple fractions. The marks are for choosing the right numbers for the top and bottom of the fraction, so it pays to slow down and count carefully, and to leave your answer as a fraction in its simplest form unless the question asks for something else.

Working out a probability

When all the outcomes are equally likely, count the ones you want and divide by the total.

  • The formula

    \(P(\text{event}) = \dfrac{\text{number of ways it can happen}}{\text{total number of equally likely outcomes}}\).

  • Example

    A bag has 3 red, 5 blue and 2 green counters. There are 10 counters, so \(P(\text{blue}) = \dfrac{5}{10} = \dfrac{1}{2}\).

  • Not happening

    \(P(\text{not } A) = 1 - P(A)\). \(P(\text{not green}) = 1 - \dfrac{2}{10} = \dfrac{8}{10} = \dfrac{4}{5}\).

  • All the outcomes add up to 1

    \(\dfrac{3}{10} + \dfrac{5}{10} + \dfrac{2}{10} = 1\), a quick check that no outcome has been missed.

A probability from a bag

A bag contains 3 red counters, 5 blue counters and 2 green counters. One counter is taken at random. Work out the probability that it is red or green.

Show the solutionHide the solution
  1. 1 Count the favourable outcomes Red or green is \(3 + 2 = 5\) counters.
  2. 2 Count all the outcomes \(3 + 5 + 2 = 10\) counters.
  3. 3 Write the probability \(\dfrac{5}{10}\).
  4. 4 Simplify \(\dfrac{1}{2}\).

Answer\(\dfrac{1}{2}\)

Mutually exclusive events

Events that cannot happen at the same time are mutually exclusive, such as getting red and getting blue from one counter.

  • Addition rule

    For mutually exclusive events, \(P(A \text{ or } B) = P(A) + P(B)\).

  • Example

    \(P(\text{red or green}) = \dfrac{3}{10} + \dfrac{2}{10} = \dfrac{1}{2}\), the same answer as counting.

  • Not mutually exclusive (Higher tier)

    If both can happen, \(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\), so the overlap is not counted twice.

  • Example

    The probability of drawing a heart or a king from a pack of 52 cards is \(\dfrac{13}{52} + \dfrac{4}{52} - \dfrac{1}{52} = \dfrac{16}{52} = \dfrac{4}{13}\).

Relative frequency and expected outcomes

When you cannot count equally likely outcomes, because a dice may be biased, use an experiment.

  • Relative frequency

    \(\dfrac{\text{number of times it happened}}{\text{total number of trials}}\). A dice is thrown 120 times and gives a 6 on 30 of them, so the relative frequency of a 6 is \(\dfrac{30}{120} = \dfrac{1}{4}\).

  • Biased or not

    A fair dice gives \(\dfrac{1}{6}\) for a 6. A result of \(\dfrac{1}{4}\) suggests the dice may be biased, especially with many trials.

  • More trials, better estimate

    Relative frequency gets closer to the true probability the more times the experiment is repeated.

  • Expected number

    Expected outcomes \(=\) probability \(\times\) number of trials. A fair dice thrown 600 times is expected to give \(\dfrac{1}{6} \times 600 = 100\) sixes.

Using relative frequency

A spinner is spun 200 times. It lands on red 74 times. Another person says the probability of red is \(\dfrac{1}{4}\). Is this likely to be correct? Work out the number of times you would expect red in 200 spins if the probability is \(\dfrac{1}{4}\).

Show the solutionHide the solution
  1. 1 Expected number \(\dfrac{1}{4} \times 200 = 50\).
  2. 2 Compare with the result The result, 74, is much larger than 50.
  3. 3 Conclude The spinner is probably biased towards red, so \(\dfrac{1}{4}\) is probably not correct.
  4. 4 Relative frequency \(\dfrac{74}{200} = \dfrac{37}{100}\) is a better estimate.

Answer50, so the result of 74 suggests the spinner may be biased

What the AQA exam gives you

AQA gives very few formulae in the exam, so most of the formulae in this lesson have to be remembered.

  • Not given

    AQA does not print the addition rule \(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\), so learn it, including why the overlap is taken away.

  • Also not given

    Probabilities add up to 1, \(P(\text{not } A) = 1 - P(A)\), expected number \(=\) probability \(\times\) number of trials, and relative frequency, so learn them all.

  • Check the wording

    Mutually exclusive events cannot happen together, so the overlap is zero and the rule becomes \(P(A) + P(B)\).

Test yourself

  1. 1

    What is \(P(\text{not } A)\)?

    Show answerHide answer

    \(1 - P(A)\).

  2. 2

    What do the probabilities of all the possible outcomes add up to?

    Show answerHide answer

    1.

  3. 3

    What is the relative frequency formula?

    Show answerHide answer

    Number of times it happened divided by the number of trials.

  4. 4

    What is the probability of a total of 7 with two dice?

    Show answerHide answer

    \(\dfrac{6}{36} = \dfrac{1}{6}\).

  5. 5

    How many sixes would you expect in 120 throws of a fair dice?

    Show answerHide answer

    \(\dfrac{1}{6} \times 120 = 20\).

Exam technique: probability

Examiners award marks for a correct fraction, so write one.

  • Count the total carefully

    Check whether the total includes the item you are choosing, or whether something has been removed.

  • Simplify at the end

    \(\dfrac{6}{36}\) can be left unsimplified for the method mark, but simplify for the answer.

  • Answers as fractions, decimals or percentages

    Do not write ratios such as 1 : 6 or words such as "1 in 6" as the answer.

  • Say why

    For biased questions, compare relative frequency with the expected probability and give a conclusion.

Summary and exam focus

  • Probability is a number from 0 to 1, found as favourable outcomes divided by all outcomes.
  • \(P(\text{not } A) = 1 - P(A)\), and the probabilities of all outcomes add up to 1.
  • Relative frequency estimates probability from an experiment, and expected outcomes are probability times trials.
  • For mutually exclusive events add the probabilities.

Exam focus

A bag contains 4 red counters and 6 blue counters. One counter is taken at random. Write down the probability that it is red. (1 mark) (1 marks)

The total number of counters is \(4 + 6 = 10\), not 6, so the probability is \(\dfrac{4}{10} = \dfrac{2}{5}\). A common mistake is to put the number of blue counters on the bottom.

Key terms

The words this lesson expects you to use. Each one is linked from the first place it appears above.

Probability
A number from 0 to 1 that measures how likely an event is.
Outcome
One possible result of an experiment.
Event
One or more outcomes that you are interested in.
Sample space
A list or table of all the possible outcomes.
Mutually exclusive
Events that cannot happen at the same time.
Relative frequency
The fraction of trials in which an event happened.
Expected frequency
The number of times an event is expected to happen, found by probability times trials.
Biased
Not fair, with some outcomes more likely than others.
Random
Chosen so that every outcome has an equal chance.

Questions and answers

15 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Write down 2 marks Easier

Ten cards are numbered 1 to 10. One card is picked at random. Work out the probability that the card shows (a) a prime number, (b) a multiple of 3. [2 marks]

Mark scheme — 2 marks available

  • (a) \(\dfrac{4}{10}\) or \(\dfrac{2}{5}\) or 0.4 — B1
  • (b) \(\dfrac{3}{10}\) or 0.3 — B1

Model answer

(a) The primes are 2, 3, 5 and 7, so \(\dfrac{4}{10} = \dfrac{2}{5}\). (b) The multiples of 3 are 3, 6 and 9, so \(\dfrac{3}{10}\).

2. Exam question Choose 3 marks Easier

Here are five words: impossible, unlikely, even chance, likely, certain. Choose the best word to describe each event. (a) A fair coin lands on heads. [1 mark] (b) A normal dice lands on 7. [1 mark] (c) A fair dice lands on a number less than 6. [1 mark]

Mark scheme — 3 marks available

  • (a) Even chance — B1
  • (b) Impossible — B1
  • (c) Likely — B1

Model answer

(a) Even chance. (b) Impossible. (c) Likely, because the probability is \(\dfrac{5}{6}\).

3. Exam question Work out 4 marks Core

Jack spins a spinner 80 times. It lands on green 28 times. (a) Work out the relative frequency of green. Give your answer as a fraction in its simplest form. [2 marks] (b) Jack spins the spinner 200 more times. Estimate the number of times it lands on green. [2 marks]

Mark scheme — 4 marks available

  • (a) \(\dfrac{28}{80}\) — M1
  • (a) \(\dfrac{7}{20}\) — A1
  • (b) \(\dfrac{7}{20} \times 200\) — M1
  • (b) 70 — A1

Model answer

(a) \(\dfrac{28}{80} = \dfrac{7}{20}\). (b) \(\dfrac{7}{20} \times 200 = 70\).

4. Exam question Work out 3 marks Core

A bag contains only red, blue and yellow counters. The probability of picking a red counter is 0.3. The probability of picking a blue counter is 0.45. There are 40 counters in the bag. Work out the number of yellow counters. [3 marks]

Mark scheme — 3 marks available

  • \(1 - 0.3 - 0.45\) — M1
  • \(0.25\) — A1
  • 10 — A1

Model answer

\(P(\text{yellow}) = 1 - 0.3 - 0.45 = 0.25\). Then \(0.25 \times 40 = 10\) yellow counters.

5. Exam question Work out 2 marks Easier

The probability that a seed grows is 0.9. Ella plants 300 seeds. Work out an estimate for the number of seeds that grow. [2 marks]

Mark scheme — 2 marks available

  • \(0.9 \times 300\) — M1
  • 270 — A1

Model answer

\(0.9 \times 300 = 270\).

6. Exam question Work out 3 marks Stretch

\(A\) and \(B\) are events. \(P(A) = 0.5\), \(P(B) = 0.4\) and \(P(A \text{ or } B) = 0.7\). Work out \(P(A \text{ and } B)\). [3 marks]

Mark scheme — 3 marks available

  • \(0.5 + 0.4 = 0.9\) — M1
  • \(0.9 - 0.7\) — M1
  • \(0.2\) — A1

Model answer

\(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\), so \(0.7 = 0.9 - P(A \text{ and } B)\). Then \(P(A \text{ and } B) = 0.9 - 0.7 = 0.2\).

7. Multiple choice 1 mark Easier

What is the probability of an impossible event?

  1. A \(1\)
  2. B \(0\) Correct
  3. C \(\dfrac{1}{2}\)
  4. D \(-1\)

Why: An impossible event has probability 0.

8. Multiple choice 1 mark Easier

A bag has 3 red, 5 blue and 2 green counters. One is taken at random. What is the probability it is blue?

  1. A \(\dfrac{1}{2}\) Correct
  2. B \(\dfrac{1}{5}\)
  3. C \(\dfrac{1}{3}\)
  4. D \(\dfrac{3}{5}\)

Why: There are 10 counters and 5 are blue, so \(\dfrac{5}{10} = \dfrac{1}{2}\).

9. Multiple choice 1 mark Core

A bag has 3 red, 5 blue and 2 green counters. What is the probability that a counter taken at random is not green?

  1. A \(\dfrac{1}{5}\)
  2. B \(\dfrac{2}{5}\)
  3. C \(\dfrac{3}{10}\)
  4. D \(\dfrac{4}{5}\) Correct

Why: \(1 - \dfrac{2}{10} = \dfrac{8}{10} = \dfrac{4}{5}\).

10. Multiple choice 1 mark Easier

A dice is thrown 120 times and a 6 comes up 30 times. What is the relative frequency of a 6?

  1. A \(\dfrac{1}{6}\)
  2. B \(\dfrac{1}{30}\)
  3. C \(\dfrac{1}{4}\) Correct
  4. D \(4\)

Why: \(\dfrac{30}{120} = \dfrac{1}{4}\).

11. Multiple choice 1 mark Core

A fair dice is thrown 600 times. How many sixes are expected?

  1. A \(60\)
  2. B \(100\) Correct
  3. C \(120\)
  4. D \(600\)

Why: \(\dfrac{1}{6} \times 600 = 100\).

12. Multiple choice 1 mark Core

Two fair dice are thrown. What is the probability that the total score is 7?

  1. A \(\dfrac{1}{6}\) Correct
  2. B \(\dfrac{1}{12}\)
  3. C \(\dfrac{7}{36}\)
  4. D \(\dfrac{1}{36}\)

Why: There are 6 ways to make 7 out of 36, so \(\dfrac{6}{36} = \dfrac{1}{6}\).

13. Multiple choice 1 mark Core

A bag has 3 red, 5 blue and 2 green counters. What is the probability of taking a red or a green counter?

  1. A \(\dfrac{3}{5}\)
  2. B \(\dfrac{1}{5}\)
  3. C \(\dfrac{3}{10}\)
  4. D \(\dfrac{1}{2}\) Correct

Why: The events are mutually exclusive, so \(\dfrac{3}{10} + \dfrac{2}{10} = \dfrac{1}{2}\).

14. Multiple choice 1 mark Core

A spinner is spun 200 times and lands on red 74 times. How many reds would be expected if the probability of red were \(\dfrac{1}{4}\)?

  1. A \(74\)
  2. B \(25\)
  3. C \(50\) Correct
  4. D \(200\)

Why: \(\dfrac{1}{4} \times 200 = 50\). The result of 74 suggests the spinner may be biased.

15. Multiple choice 1 mark Stretch

A card is taken from a pack of 52. What is the probability that it is a heart or a king?

  1. A \(\dfrac{17}{52}\)
  2. B \(\dfrac{4}{13}\) Correct
  3. C \(\dfrac{1}{13}\)
  4. D \(\dfrac{5}{26}\)

Why: \(\dfrac{13}{52} + \dfrac{4}{52} - \dfrac{1}{52} = \dfrac{16}{52} = \dfrac{4}{13}\).