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Probability

Probability

Conditional probability

Tree Diagram And Rules Extended 6 min read

On this page
  1. In this note
  2. Formula bank
  3. 'Given that' shrinks the total
  4. Two-way tables
  5. Conditional probability on a tree
  6. Ways they test it
  7. Mistakes to avoid
  8. Summary
  9. Practice
  10. Can I do this?
  11. Quiz

In this note

Syllabus E8.4: conditional probability
  • E8.4Understand 'given that': the probability inside a smaller group.
  • E8.4Find conditional probabilities from Venn diagrams and two-way tables.
  • E8.4Find conditional probabilities from tree diagrams.

Before you start: Sample spaces and Venn diagrams and Tree diagrams.

Formula bank

Given thatLearn it\(P(\text{A given B})=\dfrac{\text{number in A and B}}{\text{number in B}}\)the 'given' group becomes the new total.Use it when 'given that', 'of those who…', 'if … what is the probability'.
With probabilitiesLearn it\(P(\text{A given B})=\dfrac{P(A\cap B)}{P(B)}\)same idea with probabilities instead of counts. You don't need the notation P(A|B).Use it when Venn or tree with decimals.

'Given that' shrinks the total

40 students: football and cricketξFC16699
Given they play football: only the 22 inside F count.

'Given that a student plays football' means you already know they are in circle F. So the total is no longer 40 but the 22 in F. Of those, 6 also play cricket: \(P(\text{cricket given football})=\tfrac{6}{22}=\tfrac{3}{11}\).

General methodHow to answer any conditional probability question

  1. Find the 'given' group. Its size is the NEW total.
  2. Inside that group, count the ones you want.
  3. Probability = wanted ÷ new total.
  4. With probabilities instead of counts: divide P(both) by P(given).

Try it yourself

1
From the same diagram: given a student plays cricket, P(they also play football)? (fraction)
Show answer

Answer 2/5

Two-way tables

GlassesNo glassesTotal
Boys61420
Girls92130
Total153550

Example A student is picked at random. Given that the student is a girl, find the probability she wears glasses.

Given: girls only, so the total is 30.

  1. \(\tfrac{9}{30}\)

Answer \(\tfrac{3}{10}\)

Example Given that the student wears glasses, find the probability it is a girl.

Given: glasses only, total 15.

  1. \(\tfrac{9}{15}\)

Answer \(\tfrac35\)

Try it yourself

1
Given the student is a boy, P(no glasses)? (fraction)
Show answer

Answer 7/10

Conditional probability on a tree

Tree diagram: weather then whether the bus is lateWeatherBusRain0.3Late0.4On time0.6Dry0.7Late0.1On time0.9
The two highlighted routes end in 'Late'.

P(rain) = 0.3. If it rains the bus is late with probability 0.4; if dry, 0.1. The second branches already ARE conditional ('given rain'). To go backwards, 'given the bus was late, what is the probability it rained?', the late routes become the new total.

Example Given that the bus was late, find the probability that it rained.

New total = all late routes.

  1. \(P(\text{late})=0.3\times0.4+0.7\times0.1=0.19\)
  2. \(P(\text{rain and late})=0.12\)
  3. \(\tfrac{0.12}{0.19}\)

Answer \(\tfrac{12}{19}\approx0.632\)

Ways they test it

Type 1 From a Venn diagram

"A student who plays football is chosen. Find the probability they also play cricket." (2 marks)

Spot it 'who plays…', 'given that'.

  1. Total = that circle.
  2. Overlap ÷ circle.

For example 22 football, 6 both

  1. \(\tfrac{6}{22}\)

Answer \(\tfrac{3}{11}\)

Type 2 From a table

"Given that the student is a girl…" (1-2 marks)

Spot it a two-way table.

  1. Use the girls' row total.

Type 3 From a tree

"Given that the bus was late, find…" (3 marks)

Spot it 'given' after the event.

  1. Add the routes for the given event.
  2. Wanted route ÷ that sum.

Mistakes to avoid

  • ✗ Dividing by the grand total✓ divide by the 'given' groupThat is the whole point of 'given'.
  • ✗ Mixing the order: girl-given-glasses vs glasses-given-girl✓ the group AFTER 'given' is the total
  • ✗ Multiplying again on a tree for 'given'✓ divide by the given event's probability

Summary

  • 'Given that' = the given group is the new total.
  • Venn: overlap ÷ the given circle. Table: cell ÷ its row or column total.
  • Tree backwards: wanted route ÷ sum of the given routes.

If you see …

  • 'given that', 'of those'shrink the total.

Practice

Practice: easy, medium, hard

Easy

1
Venn: 22 play football, 6 of them cricket. P(cricket given football)? (fraction)
Show answer

Answer 3/11

2
Table: given a girl, P(glasses)? (fraction)
Show answer

Answer 3/10

3
Table: given glasses, P(girl)? (fraction)
Show answer

Answer 3/5

Medium

1
P(A) = 0.5, P(A ∩ B) = 0.15. P(B given A)?
Guide me

What is it asking? a probability

What do we know? given A: total 0.5

Plan 0.15 ÷ 0.5

Show working
  1. 0.3

Answer 0.3

2
P(B) = 0.4, P(A ∩ B) = 0.15. P(A given B)? (decimal)
Guide me

What is it asking? a probability

What do we know? given B: total 0.4

Plan 0.15 ÷ 0.4

Show working
  1. 0.375

Answer 0.375

3
Table: given glasses, P(boy)? (fraction)
Guide me

What is it asking? a probability

What do we know? glasses total 15

Plan 6 boys

Show working
  1. \(\tfrac{6}{15}\)

Answer 2/5

Hard

1
Bus tree: P(late)?
Guide me

What is it asking? a probability

What do we know? two late routes

Plan 0.12 + 0.07

Show working
  1. 0.19

Answer 0.19

PartAnswer / workingMarksGuidance
0.3 × 0.4 + 0.7 × 0.1M1
0.19A1
Total: 2 marks
2
Given the bus was late, P(rain)? (fraction or 3 d.p.)
Guide me

What is it asking? a probability

What do we know? rain and late = 0.12

Plan ÷ 0.19

Show working
  1. \(\tfrac{12}{19}\)

Answer 12/19

PartAnswer / workingMarksGuidance
0.12 / 0.19M1
12/19A1
Total: 2 marks
3
Venn (football 16 only, both 6, cricket 9 only, neither 9): given a student plays at least one sport, P(both)? (fraction)
Guide me

What is it asking? a probability

What do we know? at least one sport: 31

Plan both: 6

Show working
  1. \(\tfrac{6}{31}\)

Answer 6/31

PartAnswer / workingMarksGuidance
31 as totalM1
6/31A1
Total: 2 marks

Can I do this?

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

Quiz

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