Is P(B\mid A)=P(A\mid B) in general?

No. They have different denominators, P(A) and P(B).

What do you do along the branches of a tree diagram?

Multiply along branches; add the results from different routes.

Why do probabilities change in a draw without replacement?

The number of items and the number of each type left both change.

When are events independent?

When P(A\cap B)=P(A)P(B), equivalently P(B\mid A)=P(B).

How do you test whether two events are independent?

Check whether P(A)\times P(B)=P(A\cap B).

What does mutually exclusive mean?

P(A\cap B)=0; the events cannot occur together.

Can events with non-zero probability be both independent and mutually exclusive?

No. Exclusive events have P(A\cap B)=0 but independent events with non-zero probability have P(A\cap B)=P(A)P(B)0.

Why can a positive test for a rare disease still be unlikely to mean disease?

Most positives are false positives from the large healthy group.

Conditional probability and independenceEdexcel International A Level Maths: Flashcards

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Definition of $P(B\mid A)$?

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Definition of $P(B\mid A)$ ?: $P(B\mid A)=\frac{P(A\cap B)}{P(A)}$ , the probability of $B$ given that $A$ has occurred.
Multiplication rule?: $P(A\cap B)=P(A)P(B\mid A)$
Is $P(B\mid A)=P(A\mid B)$ in general?: No. They have different denominators, $P(A)$ and $P(B)$ .
What do you do along the branches of a tree diagram?: Multiply along branches; add the results from different routes.
Why do probabilities change in a draw without replacement?: The number of items and the number of each type left both change.
When are events independent?: When $P(A\cap B)=P(A)P(B)$ , equivalently $P(B\mid A)=P(B)$ .
How do you test whether two events are independent?: Check whether $P(A)\times P(B)=P(A\cap B)$ .
Independent: $P(A\mid B)=$ ?: $P(A)$
What does mutually exclusive mean?: $P(A\cap B)=0$ ; the events cannot occur together.
Can events with non-zero probability be both independent and mutually exclusive?: No. Exclusive events have $P(A\cap B)=0$ but independent events with non-zero probability have $P(A\cap B)=P(A)P(B)>0$ .
Addition rule for independent events?: $P(A\cup B)=P(A)+P(B)-P(A)P(B)$
If $A$ and $B$ are independent, what is $P(A\cap B')$ ?: $P(A)P(B')$
Why can a positive test for a rare disease still be unlikely to mean disease?: Most positives are false positives from the large healthy group.

Exam questions on Conditional probability and independence

A bag contains 5 red counters and 3 blue counters. Two counters are drawn at random, one after the other, without replacement.
Find the probability that the two counters are of different colours.2 marks
Events $A$ and $B$ are such that $P(A)=0.5$ , $P(B)=0.4$ and $P(A\cap B)=0.2$ .
Find $P(A\mid B')$ .2 marks
Events $A$ and $B$ are independent, with $P(A)=0.3$ and $P(A\cup B)=0.58$ .
Find $P(B)$ .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).