Sheet № 72 · Higher only · AQA · Edexcel · OCR
Conditional Probability –
Conditional probability is one of the most challenging Higher-tier topics in GCSE Maths, yet it is tested regularly on AQA, Edexcel and OCR papers. It deals with the probability of an event happening given that another event has already occurred. This changes the sample space and often changes the probability itself. Questions typically i
§Key definitions
Conditional probability
is the probability of event B happening given that event A has already happened. It is written as P(B | A), read as "the probability of B given A".
Question:
A bag contains 7 blue and 3 red counters. Two counters are taken without replacement. Find (a) the probability that both are blue, (b) the probability that the second counter is red given that the first was blue.
(a)
First pick: P(blue) = 7/10. After removing one blue, there are 6 blue and 3 red left (9 total). Second pick: P(blue | first blue) = 6/9 = 2/3.
(b)
Given the first counter is blue, there are 6 blue and 3 red remaining (9 counters).
Question 1:
A box contains 5 milk and 3 dark chocolates. Two are chosen without replacement. Find the probability that both are dark.
§Formulas to memorise
P(B | A) = P(A ∩ B) ÷ P(A)
Independent events: — The outcome of one does not affect the other. P(B | A) = P(B). Example: flipping a coin twice.
Dependent events: — The outcome of the first affects the probabilities for the second. P(B | A) ≠ P(B). Example: picking two cards from a deck without replacement.
Conditional probability: is the probability of event B happening given that event A has already happened. It is written as P(B | A), read as "the probability of B given A".
Divide: P(B | A) = n(A ∩ B) ÷ n(A).
Divide: P(B | A) = P(A ∩ B) ÷ P(A).
Worked example
A bag contains 7 blue and 3 red counters. Two counters are taken without replacement. Find (a) the probability that both are blue, (b) the probability that the second counter is red given that the first was blue.
Working:
⚠ Common mistakes
- ✗Forgetting to reduce the total — in "without replacement" problems, the denominator decreases. If you start with 10 items, the second pick has 9 items, not 10.
- ✗Using the whole sample space instead of the restricted one — conditional probability means you only consider the subset where the given event has occurred.
- ✗Confusing P(A ∩ B) with P(B | A) — the intersection is the probability of both; the conditional is the probability of one given the other. They are related by the formula but are not the same.
- ✗Assuming independence — do not assume events are independent unless the question tells you they are. Check by comparing P(B | A) with P(B).
- ✗Misidentifying which event is "given" — P(A | B) ≠ P(B | A). Read the question carefully to see which event has already occurred.
✦ Exam tips
- →Read the question twice — identify which event is given and which event you are finding the probability of. The word "given" or the phrase "given that" is your signal.
- →Draw a tree diagram for without-replacement problems — it makes the adjusted probabilities visible and reduces errors.
- →Two-way table shortcut — when given a table, conditional probability is simply "look at the row/column for the condition, then read the relevant cell". No formula needed if you understand this logic.
- →Show the formula — even if you can see the answer, write P(B | A) = P(A ∩ B) ÷ P(A) to demonstrate understanding and earn method marks.
- →Testing for independence — if P(B | A) = P(B), the events are independent. If not, they are dependent. State this conclusion clearly.