POP

E8.4 Conditional Probability: Venn Diagrams, Tables & Trees

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Conditional probability asks: given that something is already known to be true, what is the probability of something else? Formal $P(A\mid B)$ notation is not required by this syllabus, but the underlying idea — restricting to a smaller sample space — is essential.

What does "given that..." actually do to a calculation?

It shrinks the "total" you divide by, down to only the outcomes that satisfy the stated condition:

P(\text{event given condition}) = \frac{\text{outcomes satisfying both}}{\text{outcomes satisfying the condition}}

How do I find a conditional probability from a two-way table?

Read across the row (or down the column) that matches the given condition, and use that row/column's own total as the new denominator — not the grand total of the whole table.

	Plays an Instrument	No Instrument	Total
Boys	$14$	$26$	$40$
Girls	$22$	$18$	$40$
Total	$36$	$44$	$80$

Given that a student is a girl, find the probability she plays an instrument: the condition "girl" restricts the total to the girls' row ( $40$ ), of which $22$ play an instrument:

P(\text{instrument}\mid\text{girl}) = \frac{22}{40} = \frac{11}{20}

Exam Tip

Writing this as $\dfrac{22}{80}$ instead of $\dfrac{22}{40}$ is the most common error in this topic — the word "given" always restricts you to a smaller total, never the grand total.

Given that a student plays an instrument, find the probability they are a boy: now the condition restricts to the instrument column ( $36$ ), of which $14$ are boys:

P(\text{boy}\mid\text{instrument}) = \frac{14}{36} = \frac{7}{18}

Notice this used a completely different denominator ( $36$ , not $40$ ) — the event named after "given" is always what fixes the new total.

How do I find a conditional probability from a Venn diagram?

Identify the region representing the given condition, use only the values inside that region as your new total, then find what fraction of it also satisfies the second event.

Worked Example

In a survey of $60$ people, $35$ like coffee, $28$ like tea, and $15$ like both. A person is chosen at random.

(a) Given that the person likes tea, find the probability they also like coffee.

(b) Given that the person does not like tea, find the probability they like coffee.

Number who like tea only: $28-15=13$ . Number who like coffee only: $35-15=20$ . Number who like neither: $60-(13+15+20)=12$ .
(a) The condition "likes tea" restricts the total to everyone in the tea circle: $13+15=28$ . Of these, $15$ also like coffee:

P(\text{coffee}\mid\text{tea}) = \frac{15}{28}

(b) The condition "does not like tea" restricts the total to everyone outside the tea circle: $60-28=32$ . Of these, only the "coffee only" group of $20$ also likes coffee:

P(\text{coffee}\mid\text{not tea}) = \frac{20}{32} = \frac{5}{8}

Exam Tip

Before dividing, always re-draw or re-label the Venn diagram's regions with actual numbers first — trying to read conditional probabilities directly off percentages or unlabelled circles is where most marks are lost.