What does P(A|B) mean?

The probability of A given that B has occurred.

Is P(A|B) always equal to P(B|A)?

No, they are different unless P(A)=P(B).

How do the second-stage probabilities change without replacement?

They change, because the counts and the total have reduced.

How do you find P(revised|pass) from a tree?

\frac{P(revised and pass)}{P(pass)}, with P(pass) from all routes.

How do you use a two-way table for P(A|B)?

Divide the count in both A and B by the total in B.

Conditional probabilityAQA A-Level Maths: Flashcards

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Question

State the conditional probability formula.

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State the conditional probability formula.: $P(A|B)=\frac{P(A\cap B)}{P(B)}$
What does $P(A|B)$ mean?: The probability of $A$ given that $B$ has occurred.
Rearrange the formula to give $P(A\cap B)$ .: $P(A\cap B)=P(B)\times P(A|B)$
In $P(A|B)$ , which event is the denominator based on?: $B$ , the condition.
Is $P(A|B)$ always equal to $P(B|A)$ ?: No, they are different unless $P(A)=P(B)$ .
What is the independence test using conditional probability?: $P(A|B)=P(A)$ .
What is the independence test using the multiplication rule?: $P(A\cap B)=P(A)P(B)$ .
On a tree diagram, what do you do along a route?: Multiply the probabilities.
On a tree diagram, what do you do for different routes to one event?: Add the route probabilities.
How do the second-stage probabilities change without replacement?: They change, because the counts and the total have reduced.
How do you find $P(\text{revised}|\text{pass})$ from a tree?: $\frac{P(\text{revised and pass})}{P(\text{pass})}$ , with $P(\text{pass})$ from all routes.
Bag: 5 red, 3 blue, no replacement. $P(\text{blue second}|\text{blue first})$ ?: $\frac27$
How do you use a two-way table for $P(A|B)$ ?: Divide the count in both $A$ and $B$ by the total in $B$ .

Exam questions on Conditional probability

In a group of 120 students, 70 study Biology, 50 study Chemistry and 20 study both. Twenty students study neither subject. One student is chosen at random.
Determine whether studying Biology and studying Chemistry are independent.2 marks
A bag contains 5 red counters and 3 blue counters. Two counters are taken at random, one after the other, without replacement.
Given that the second counter is red, find the probability that the first counter was blue.2 marks
In a survey of 80 people, 45 own a bicycle, 30 own a car and 12 own both.
Find the probability that a person chosen at random owns neither a bicycle nor a car. Given that they own at least one of the two, find the probability that they own a bicycle.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).