Lesson 10 of 10 · 12 min

Chapter review

Must-know facts

13 facts

1P(E|F) = P(E ∩ F)/P(F), with P(F) ≠ 0; for equally likely outcomes it is n(E ∩ F)/n(F).
2P(E|F) and P(F|E) are different in general.
3P(S|F) = P(F|F) = 1 and P(E′|F) = 1 − P(E|F).
4Union under a condition: P((A ∪ B)|F) is P(A|F) plus P(B|F) minus P((A ∩ B)|F).
5Multiplication rule: P(E ∩ F) = P(E)P(F|E), and equally P(F)P(E|F).
6Three events: P(A ∩ B ∩ C) = P(A)P(B|A)P(C|A ∩ B).
7E and F are independent exactly when P(E ∩ F) = P(E)P(F).
8Independent and mutually exclusive are different ideas; with non-zero probabilities, no pair of events is both.
9If E and F are independent, so are E and F′, E′ and F, E′ and F′.
10For independent A and B, P(A ∪ B) = 1 − P(A′)P(B′).
11A partition is pairwise disjoint, exhaustive, with every part of non-zero probability.
12Total probability: P(A) = Σ P(Eⱼ)P(A|Eⱼ) over a partition.
13Bayes' theorem gives the posterior: P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ) ÷ P(A), with P(A) from total probability.

Common traps

Where marks are lost

Reading P(A|B) as P(B|A).

The event after the bar is the one known to have occurred; it supplies the denominator.

Writing P(E ∩ F) = P(E)P(F) without checking independence.

In general P(E ∩ F) = P(E)P(F|E). Drop the condition only when the events are independent.

Treating draws without replacement as if the counts stayed the same.

Reduce the count and the total after each draw: 10/15, then 9/14.

Calling mutually exclusive events independent.

Mutually exclusive means P(E ∩ F) = 0; independent means P(E ∩ F) = P(E)P(F).

Taking P(E|F′) = 1 − P(E|F).

The complement rule applies to the event: P(E′|F) = 1 − P(E|F).

Counting outcomes when they are not equally likely.

Use P(E ∩ F)/P(F) with the actual probabilities, as in the coin-then-die experiment.

Leaving a hypothesis out of the denominator in Bayes' theorem.

The denominator is P(A): one product for every part of the partition.

Checking only the three pairs for mutual independence of A, B, C.

The triple product must hold as well: P(A ∩ B ∩ C) equals P(A)P(B)P(C).

Formulas

8 to know

Conditional probability

P(E|F) = P(E ∩ F) / P(F)

P(F) ≠ 0.

Complement under a condition

P(E′|F) = 1 − P(E|F)

Same condition F on both sides.

Multiplication rule

P(E ∩ F) = P(E) P(F|E)

Also equals P(F) P(E|F).

Independent events

P(E ∩ F) = P(E) P(F)

Holds exactly when E and F are independent.

At least one of two independent events

P(A ∪ B) = 1 − P(A′) P(B′)

A, B independent.

Total probability

P(A) = Σ P(Eⱼ) P(A|Eⱼ)

E₁, …, Eₙ a partition of S.

Bayes' theorem

P(Eᵢ|A) = P(Eᵢ) P(A|Eᵢ) ÷ P(A)

P(A) = Σ P(Eⱼ) P(A|Eⱼ), the total over the partition.

Mean of a random variable

E(X) = Σ xᵢ pᵢ

JEE extension.

Key terms

7 terms

Conditional probability: The probability of E computed inside F, once F is known to have occurred.
Independent events: Events for which the occurrence of one leaves the probability of the other unchanged.
Mutually exclusive events: Events with no outcome in common.
Partition: Events that are pairwise disjoint, cover the sample space, and each have non-zero probability.
Prior probability: P(Eᵢ), the probability of a hypothesis before the evidence is seen.
Posterior probability: P(Eᵢ|A), the probability of a hypothesis after the evidence A is seen.
Random variable: A real-valued function on the sample space of a random experiment.