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Probability

Probability

Probability of an event

Concept And Formula Core + Extended 7 min read

On this page
  1. In this note
  2. Formula bank
  3. The probability of one event
  4. Relative and expected frequency
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 8.1 and 8.2: probability of a single event
  • 8.1Use the probability scale from 0 to 1 and the notation P(A) and P(A′).
  • 8.1Find the probability of a single event and of it not happening.
  • 8.2Use relative frequency and expected frequency; know fair, biased and random.

Before you start: fractions, decimals and percentages. Next: Sample spaces and Venn diagrams.

Formula bank

Single eventLearn it\(P(A)=\dfrac{\text{number of ways A can happen}}{\text{total number of outcomes}}\)outcomes must be equally likely.Use it when picking at random.
Not happeningLearn it\(P(A')=1-P(A)\)Use it when 'not', 'does not'.
All outcomesLearn itprobabilities of all outcomes add to 1Use it when a table with a missing probability.
Relative frequencyLearn it\(\dfrac{\text{number of times it happened}}{\text{number of trials}}\)an ESTIMATE of the probability.Use it when results of an experiment.
Expected frequencyLearn it\(P(A)\times\text{number of trials}\)Use it when 'how many times would you expect'.

The probability of one event

Probability measures how likely something is, from 0 (impossible) to 1 (certain). Write it as a fraction, decimal or percentage, never as '3 in 10' or '3 : 10'. A bag has 5 red, 3 blue and 2 green counters: \(P(\text{red})=\tfrac{5}{10}=\tfrac12\).

General methodHow to answer any single-event question

  1. Count the total number of equally likely outcomes.
  2. Count the ones you want.
  3. Probability = wanted ÷ total. Simplify the fraction if you like.
  4. 'Not' something? Do 1 − P(it).

Example The bag has 5 red, 3 blue and 2 green counters. Find P(not green).

Use 1 − P(green).

  1. \(P(\text{green})=\tfrac{2}{10}\)
  2. \(1-\tfrac{2}{10}=\tfrac{8}{10}\)

Answer \(\tfrac45\)

Example A spinner can land on A, B, C or D. P(A) = 0.2, P(B) = 0.35, P(C) = x and P(D) = 2x. Find x.

All probabilities add to 1.

  1. \(0.2+0.35+x+2x=1\)
  2. \(3x=0.45\)
  3. \(x=0.15\)

Answer x = 0.15

Try it yourself

1
P(a green pen) = 0.17. P(not a green pen)?
Show answer

Answer 0.83

2
P(win) = 0.35 and P(draw) = 0.4. P(lose)?
Show answer

Answer 0.25

Relative and expected frequency

When you can't count outcomes (a biased spinner, a drawing pin), do an experiment. The relative frequency = how often it happened ÷ number of trials. More trials give a better estimate. A spinner landing on 1 in 46 of 200 spins: relative frequency \(\tfrac{46}{200}=0.23\).

Fair means every outcome is equally likely; biased means it isn't; at random means every item has the same chance of being picked.

General methodHow to find an expected frequency

  1. Find (or estimate) the probability.
  2. Multiply by the number of trials (or people).
  3. The answer is an estimate; it does not have to happen exactly.

Example The probability of rain on a day in June is 0.3. On how many of the 30 days would you expect rain?

P × trials.

  1. \(0.3\times30=9\)

Answer 9 days

Example In a survey, 64 of 400 people used a voucher. Estimate how many of 1000 people would use it.

Relative frequency first.

  1. \(\tfrac{64}{400}=0.16\)
  2. \(0.16\times1000=160\)

Answer 160 people

Try it yourself

1
A fair dice is rolled 120 times. Expected number of sixes?
Show answer

Answer 20

2
18 of 50 spins land on red. Estimate how many of 75 spins land on red.
Show answer

Answer 27

Ways they test it

Type 1 Single event

"A letter is picked at random from PROBABILITY. Find P(B)." (1 mark)

Spot it 'at random', one pick.

  1. Wanted ÷ total.

Type 2 Not

"The probability it rains is 0.17. Find the probability it does not rain." (1 mark)

Spot it 'not'.

  1. 1 − P.

Type 3 Missing probability

"Complete the table." or "Find x." (2 marks)

Spot it a table of probabilities with a gap or x.

  1. Everything adds to 1.
  2. Solve for x.

For example 0.2, 0.35, x, 2x

  1. \(3x=0.45\)

Answer x = 0.15

Type 4 Expected / relative frequency

"How many times would you expect…?" (1-2 marks)

Spot it a number of trials.

  1. P × trials.
  2. From results: relative frequency first.

Mistakes to avoid

  • ✗ Writing '3 in 10' or 3 : 10✓ \(\tfrac{3}{10}\), 0.3 or 30%Ratios and words score zero.
  • ✗ A probability bigger than 1✓ check: every probability is between 0 and 1
  • ✗ Expected frequency as a probability✓ it is a NUMBER of times

Summary

  • P = wanted ÷ total equally likely outcomes.
  • P(not A) = 1 − P(A); all outcomes add to 1.
  • Relative frequency estimates P; expected frequency = P × trials.

If you see …

  • 'not'1 − P.
  • 'how many would you expect'P × number.

Practice

Practice: easy, medium, hard

Easy

1
5 red, 3 blue, 2 green counters. P(red)? (fraction or decimal)
Show answer

Answer 1/2

2
Same bag. P(blue)?
Show answer

Answer 3/10

3
46 of 200 spins land on 1. Relative frequency?
Show answer

Answer 0.23

Medium

1
P(A) = 0.2, P(B) = 0.35, P(C) = x, P(D) = 2x. P(D)?
Guide me

What is it asking? a probability

What do we know? 3x = 0.45

Plan x = 0.15, so 2x

Show working
  1. 0.3

Answer 0.3

2
P(rain) = 0.3. Expected rainy days in 30?
Guide me

What is it asking? a number

What do we know? P × days

Plan 0.3 × 30

Show working
  1. 9

Answer 9

3
7 out of 25 people use a voucher. How many of 500 would you expect?
Guide me

What is it asking? a number

What do we know? 7/25 = 0.28

Plan × 500

Show working
  1. 140

Answer 140

Hard

1
Four outcomes have probabilities 0.1, 0.25, 0.3 and y. Find y.
Guide me

What is it asking? a probability

What do we know? add to 1

Plan 1 − 0.65

Show working
  1. 0.35

Answer 0.35

PartAnswer / workingMarksGuidance
1 − (0.1 + 0.25 + 0.3)M1
0.35A1
Total: 2 marks
2
64 of 400 people used a voucher. How many of 1000 would you expect?
Guide me

What is it asking? a number

What do we know? relative frequency 0.16

Plan × 1000

Show working
  1. 160

Answer 160

PartAnswer / workingMarksGuidance
64/400 × 1000M1
160A1
Total: 2 marks
3
18 of 50 spins land on red. How many of 75 would you expect?
Guide me

What is it asking? a number

What do we know? 18/50 = 0.36

Plan × 75

Show working
  1. 27

Answer 27

PartAnswer / workingMarksGuidance
0.36 × 75M1
27A1
Total: 2 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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