Conditional probability

Find P(A | B), or use it to find P(A ∩ B)

In 8 of 20 recent exams
Paper 1 or 2
2 to 4 marks

How it is set in the exam

Paper 1 (no calculator) mostly (6 exam versions have a Paper 1 (no calculator) part, 3 a Paper 2 (calculator) part), 2–4 marks, nearly always the last part after a Venn diagram or a tree has been filled. Common (8 of 20 exam versions). A tree is the tool in 4 of them, and 3 of those are reversals ("given the result, which bag?").

Spot it in the exam

Exam phrases to spot (5)

“given that”
“find $\mathrm{P}(A \mid B)$ ”
“find $\mathrm{P}(A \mid B')$ ”
“given that the ball is white, find the probability that it came from bag $A$ ”
“given $\mathrm{P}(A \mid B)$ , find $\mathrm{P}(A \cap B)$ ”

In disguise (3)

a conditional stated in words as a fraction of a group: "of the students who study Spanish, $\frac{3}{4}$ also study music" is $\mathrm{P}(M \mid S) = \frac{3}{4}$ , not $\mathrm{P}(M \cap S)$
the result is known and the question asks about the first stage, with no "given that": "a plant is taller than 40 cm; find the probability that it came from grower Q"
"what proportion of the … are …": one row or column of a table is the reduced sample space

Variants (3)

tool: the formula alone, a Venn diagram, a two-way table, or a tree (read forwards or reversed)
condition: an event or its complement ( $\mathrm{P}(A \mid B')$ )
asked: the conditional, or the intersection from a given conditional (the multiplication form)

Looks like it, but isn't (4)

both events are ranges of one numerical variable (" $X > a$ ", "over 60 years old", "at least 3 successes"): Given that X is in a range. Draw both ranges on a number line.
two formulas share an unknown and combine into an equation: Probabilities in terms of an unknown
the events are independent and a conditional is asked: it is just $\mathrm{P}(A)$ (Independent events)
a fraction of a group is a branch, and a total over the groups is asked: Tree diagrams and repeated draws

Worked example

The condition is a complement, from a Venn diagram

Paper 1 (no calculator)

Question

Events $A$ and $B$ are such that $\mathrm{P}(A) = 0.46$ and $\mathrm{P}(B) = 0.74$ , and the probability that neither $A$ nor $B$ occurs is $0.12$ .
(a) Find $\mathrm{P}(A \cap B)$ . [3]
(b) Find $\mathrm{P}(A \mid B')$ . [3]
(c) Find $\mathrm{P}(B \mid A')$ . [2]

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Which method?

Venn diagram
The intersection $\mathrm{P}(A \cap B)$ is
the overlap region; for $\mathrm{P}(A \mid B')$ , the part of $A$ outside $B$
The condition $\mathrm{P}(B)$ is
the whole of $B$ ; for $B'$ , $1 - \mathrm{P}(B)$
two-way table
The intersection $\mathrm{P}(A \cap B)$ is
the cell
The condition $\mathrm{P}(B)$ is
its row or column total
tree, reversed ("which first stage?")
The intersection $\mathrm{P}(A \cap B)$ is
the path through that first stage and the result
The condition $\mathrm{P}(B)$ is
the sum of all the paths that give the result
$\mathrm{P}(A \mid B)$ given, the intersection asked
The intersection $\mathrm{P}(A \cap B)$ is
$\mathrm{P}(A \mid B)\,\mathrm{P}(B)$ (the multiplication form)
The condition $\mathrm{P}(B)$ is
given

The tool	The intersection $\mathrm{P}(A \cap B)$ is	The condition $\mathrm{P}(B)$ is
Venn diagram	the overlap region; for $\mathrm{P}(A \mid B')$ , the part of $A$ outside $B$	the whole of $B$ ; for $B'$ , $1 - \mathrm{P}(B)$
two-way table	the cell	its row or column total
tree, reversed ("which first stage?")	the path through that first stage and the result	the sum of all the paths that give the result
$\mathrm{P}(A \mid B)$ given, the intersection asked	$\mathrm{P}(A \mid B)\,\mathrm{P}(B)$ (the multiplication form)	given

Steps

Write the conditional in context
, with the events named: $\mathrm{P}(\text{bag } X \mid \text{gold})$ , or $\mathrm{P}(M \mid S')$ with $M$ and $S$ defined. The recognition mark needs this; a bare $\mathrm{P}(A \mid B)$ , or the general formula in $A$ and $B$ with neither the question's events nor its values, earns nothing.
Find the intersection and the condition
with the table above. The condition is usually an earlier part.
Divide,
with the values written in the fraction first (a correct top or bottom on its own can earn the method mark), then simplify: exact fractions on , 3 s.f. on .
Asked for the intersection instead?
Multiply: $\mathrm{P}(A \cap B) = \mathrm{P}(A \mid B)\,\mathrm{P}(B)$ .
Sanity check:
the answer is between $0$ and $1$ . In a reversed tree, if a first-stage option makes the result more likely than it is overall ( $\mathrm{P}(\text{gold} \mid X) > \mathrm{P}(\text{gold})$ ), the reversed probability $\mathrm{P}(X \mid \text{gold})$ is above the first-stage value $\mathrm{P}(X)$ ; if less likely, below.

Common mistakes

$\mathrm{P}(A \mid B)$ and $\mathrm{P}(B \mid A)$ swapped
the condition, after the bar, goes on the bottom.
$\mathrm{P}(A)$ on top instead of $\mathrm{P}(A \cap B)$
for $\mathrm{P}(A \mid B')$ the top is only the part of $A$ outside $B$ .
$\mathrm{P}(A \mid B') = 1 - \mathrm{P}(A \mid B)$
wrong. The complement works before the bar, $\mathrm{P}(A' \mid B) = 1 - \mathrm{P}(A \mid B)$ , never after it.

More mistakes (4)

A conditional in words read as "both"
"of those who study Spanish, $\frac{3}{4}$ study music" is $\mathrm{P}(M \mid S)$ .
A reversed tree over one path
the bottom is every path that gives the result.
Only the general formula written
unless the question's events or values are in it.
Rounding carried values early
on : keep them stored.

Easy to confuse with

Next typeTwo events

More in Probability: Venn diagrams, trees and conditional probability

Are the events independent?

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Easy to confuse with

More in Probability: Venn diagrams, trees and conditional probability