GCSE · Maths · Edexcel · Spec 1MA1 · Higher

Conditional probability

Take the same 18 marbles and you can write three different probabilities with them. The only thing that changes is what you were told first.

Maths · Probability

Two marbles, and the first one doesn't go back

Start on 'With replacement', then flip to 'Without replacement' and watch the second-stage fractions. Tap a row to light up one path.

A bag holds 6 purple and 4 green marbles. You take one, keep it out, then take another.

The marble goes back each time, so the bag never changes and the second-stage branches just repeat the first stage.

Tap a row

Tap an outcome row — the two branches on its path glow indigo and the multiplication is laid out.

Exam line: Multiply along a path: P(A and C) = P(A) × P(C given A). Tap P · P to check: 6/10 × 5/9.

Watch out: Nothing goes back, so the second stage is out of 9, not 10. The tree shows each product unsimplified: 30/90 is 1/3.

Which idea is yours?

'And' or 'given'?

A bag holds 60 marbles. 18 of them are swirled and blue. 29 are blue altogether (swirled or not). One marble is picked at random. Write S for swirled and B for blue.

Compare P(S and B) with P(S | B). Which idea is closest to what you think?

Both are 18/60. It's the same 18 swirled blue marbles either way.The top stays 18, but being told 'blue' shrinks the bottom to the 29 blue marbles, so P(S | B) = 18/29.The vertical line just means 'and', so P(S | B) is another way of writing P(S and B).

How sure are you?

I'm sureI think soI'm guessing

Both are 18/60. It's the same 18 swirled blue marbles either way.

This is a common misconception

It's tempting, because 18 really is the number on top both times. But the bottom number is the whole you're choosing from. For P(S and B) that's all 60 marbles. For P(S | B) you've been told the marble is blue, so you're choosing from the 29 blue ones only.

The top stays 18, but being told 'blue' shrinks the bottom to the 29 blue marbles, so P(S | B) = 18/29.

This matches the evidence

Yes. 18 swirled blue marbles are still the ones you want. P(S and B) = 18/60 because it's out of everything. P(S | B) = 18/29 because the given information makes the blue marbles the new whole.

The vertical line just means 'and', so P(S | B) is another way of writing P(S and B).

This is a common misconception

Easy slip: a symbol between two letters looks like it joins them. But the vertical line is read as 'given'. P(S | B) says 'the probability of S, given B', which is a different question from 'S and B', and that's why it has a different denominator.

'And' keeps all 60 marbles in play. 'Given blue' leaves only the 29 blue ones. Same 18 on top, different whole underneath.

Exam line: Same top number, different bottom number: the given information changes the whole.

Try it on a small bag

Sort the marbles, then use the counts

A small bag holds 10 marbles. Put each one in the circles it belongs to: swirled, blue, both, or neither. Then work out P(blue | swirled).

A Swirled
B Blue

Marble 1
Marble 2
Marble 3
Marble 4
Marble 5
Marble 6
Marble 7
Marble 8
Marble 9
Marble 10

Exam line: Overlap on top, the given set's total on the bottom.

Your turn to finish the method

Are swirled and blue independent?

A different bag holds 60 marbles. 12 are blue and 48 are not blue. 10 marbles are swirled: 2 of them are blue and 8 are not blue. Are 'swirled' and 'blue' independent?

Test: compare P(swirled | blue), P(swirled | not blue) and P(swirled). If being told the colour changes nothing, the events are independent.
missing step

Which line is step 2?

P(swirled | blue) = 2/12 = 1/6P(swirled | blue) = 2/60 = 1/30P(swirled | blue) = 2/10 = 1/5

Exam line: Independent means the given information changes nothing.

Two words that are not synonyms

Mutually exclusivevsIndependent

Easy to blur together. Compare them row by row.

Focus

What it means

Mutually exclusive

The events can't both happen.

Independent

Knowing one happened doesn't change the probability of the other.

The insight

Two different ideas with two different names. They are not synonyms.

P(A | B)

Mutually exclusive

0, because if B has happened, A can't have.

Independent

Equal to P(A). The given information changes nothing.

The opposite

Mutually exclusive

Events that can both happen.

Independent

Dependent: knowing one does change the probability of the other.

Exam-style practice

Write it, then mark it yourself

35% of students at a college walk to class and the rest take the bus. 20% of the students who walk are late. 10% of the students who take the bus are late. A late student is picked at random. Find the probability that this student walks to class. [4 marks]

Your answer

0 words · your answer stays on this page and is not sent anywhere.

The mark scheme

Chooses a total number of students and converts the percentages to counts, for example 1000 students: 350 walk and 650 take the bus. (1 mark)
Finds the late students in each group: 20% of 350 = 70 late walkers and 10% of 650 = 65 late bus users. (1 mark)
Uses the total number of late students, 70 + 65 = 135, as the denominator, because the student is known to be late. (1 mark)
Gives the probability as 70/135 = 14/27. (1 mark)

Model answer

Use 1000 students. 35% walk: 350. The other 650 take the bus. Late walkers: 20% of 350 = 70. Late bus users: 10% of 650 = 65. Given the student is late, the whole is all late students: 70 + 65 = 135. P(walks | late) = 70/135 = 14/27.

A table of counts works well here. Convert the percentages first, then the 'given' group's total becomes the denominator.

Exam line: Choose a representation, convert percentages to counts, and say what the denominator is.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Be told something first, and what you're choosing from gets smaller.

What you need to know

P(A | B) is the probability of A given that B has happened. Read the vertical line as 'given'.
Have a goA friend reads P(R | W) out loud as 'the probability of R and W'. Which one word should they have said instead of 'and'?
Given.
The vertical line means 'given'. 'And' would describe a different thing, P(R and W).
Being told B shrinks what you are choosing from to just the B outcomes. They become the new whole.
So P(A and B) and P(A | B) share a numerator, the outcomes in both, but have different denominators.
Have a goIn a year group of 50, 10 pupils both swim and play chess, and 20 swim in total. What is P(chess | swim)?
10/20, which is 1/2.
The top is the 10 in both groups. 'Given swim' shrinks the bottom to the 20 swimmers, not all 50.
In a two-way table, 'given' picks a row or column. In a Venn diagram it picks a set. Its total is the denominator.
On a tree, every second-stage branch is a conditional probability: this outcome, given the first one.
Without replacement, the bag shrinks. After one of 10 is taken, the second-stage denominators are 9.
Two taken at once? Treat it as one then another. The second stage goes down by 1, not 2.
Have a goTwo sweets are grabbed from a bag of 12 at the same time. A classmate says the second-stage denominator is 10. Are they right?
No. It's 11.
Treat it as one grabbed, then another. Only one sweet has gone before the second, so the denominator drops by 1, not 2.
Along a tree path, multiply: P(A and C) = P(A) × P(C given A). Each pair of second-stage branches adds to 1.
Events are independent when knowing one happened leaves the other's probability unchanged: P(A | B) = P(A). The opposite is dependent.
Mutually exclusive events can't both happen, so P(A | B) = 0. Mutually exclusive and independent are not synonyms.

The big picture

A conditional probability is the chance of something happening given that you already know something else has happened, written P(A | B). Being told B shrinks the set you are choosing from to just the B outcomes, so the denominator changes while the numerator does not. You can read it from a tree, a two-way table or a Venn diagram, it changes along the branches of a without-replacement tree, and it lets you test whether two events are independent.

Key points

1'Given' changes the whole you are choosing from, so the denominator changes and the numerator doesn't.

2P(A | B) means the probability of A given B, and the vertical line is read as 'given'.

3On a tree, every second-stage branch is a conditional probability, and without replacement its denominator is one smaller.

4Independent means P(A | B) = P(A). Mutually exclusive means P(A | B) = 0.

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Worked example

Problem

240 pupils at two schools each learn one language. German: 86 pupils at Oakfield Academy and 24 at Riverside School (110 in total). French: 74 at Oakfield Academy and 56 at Riverside School (130 in total). One pupil is picked at random. Given that the pupil learns German, find the probability that they go to Oakfield Academy.

⚠ Watch out

Leaving the second stage out of 10 when nothing was put back. If one of 10 has been taken, the next draw is out of 9, so purple given purple is 5/9, not 5/10.

🧠

Memory hook

'Given' means shrink the whole: same top, smaller bottom.

✓

Check yourself

Without looking back: in a bag of 12, one marble is taken and kept. What is the denominator of every second-stage branch? Then say what each second-stage branch is a probability 'given'.

Flashcards

(13)

What does 'given' do to the set you are choosing from?

It shrinks it to just the outcomes that meet the condition. They become the new whole, so the denominator changes.

How do you write 'the probability of A given B'?

P(A | B). The vertical line is read as 'given'.

P(A and B) and P(A | B): what is the same and what differs?

Same numerator: the outcomes in both. Different denominator: all outcomes for 'and', only the given set for P(A | B).

Which total becomes the denominator of a conditional probability?

The total of the given group: the row, column or set that meets the condition.

In a Venn diagram, what goes on the bottom of P(A | B)?

The count in set B alone, the given set. The overlap goes on top.

What is every second-stage branch of a tree?

A conditional probability: this outcome, given the first one.

One of 10 marbles is taken and not replaced. What is the second-stage denominator?

Two items are taken from a bag together. How do you work out the second stage?

Treat it as one then another. The denominator goes down by 1, not 2.

How do you find P(A and C) along a tree path?

Multiply: P(A) × P(C given A).

What must each pair of second-stage branches add up to?

How do you test whether two events are independent?

Compare P(A | B) with P(A). If the given information changes nothing, they are independent.

Are 'mutually exclusive' and 'independent' the same thing?

No. Mutually exclusive: P(A | B) = 0. Independent: P(A | B) = P(A). The opposite of independent is dependent.

A question gives percentages or decimals. What can you do before you start?

Convert them to counts or fractions, then choose a tree, a two-way table or a Venn diagram that fits the information.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Two marbles, and the first one doesn't go back

'And' or 'given'?

Sort the marbles, then use the counts

Are swirled and blue independent?

Write it, then mark it yourself

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