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Probability · Lecture 2 of 76 · 5:55

The Probability of the Difference of Two Events

Study guide

What this lecture covers

This short recitation problem shows how to prove a probability identity purely from the axioms, without appealing to intuition. Starting from two events A and B, the goal is to show that the probability of their symmetric difference (the points in exactly one of the two events) equals P(A) + P(B) - 2P(A ∩ B).

It's a good follow-up to the lecture on probability axioms: it demonstrates the habit of naming disjoint pieces of a Venn diagram and building up a proof step by step using only the additivity axiom.

Key ideas

Simplifying notation first: before manipulating a messy equation, name the disjoint regions involved (here called C, D, and E) to make the algebra manageable.
Partitioning into disjoint pieces: A splits into C (in A but not B) and E (in both); B splits into D (in B but not A) and E.
Additivity axiom as the only tool: every step in the proof rests on the fact that probabilities of disjoint sets add.
Adding and subtracting a term: inserting +P(E) - P(E) into an expression is a standard trick for connecting a derived quantity to known probabilities.
Result: P(C ∪ D) = P(A) + P(B) - 2P(A ∩ B), i.e. the probability that exactly one of A or B occurs.

Before you watch

Know the three probability axioms (non-negativity, normalization, additivity), covered in the first lecture of this course.
Be comfortable with basic set notation: union, intersection, and complement.

Check your understanding

Why can the additivity axiom be applied to C and D, but not directly to A and B?
How does writing A as the union of C and E help connect P(A) to P(C) and P(E)?
Why is adding and subtracting P(E) a valid algebraic step, and what does it accomplish here?
Can you sketch a Venn diagram and label the regions C, D, and E yourself before checking the derivation?

Vocabulary

symmetric difference (noun): The set of elements that belong to exactly one of two sets, not both.
The symmetric difference of A and B excludes their overlap.
notation (noun): A system of symbols used to represent mathematical ideas clearly.
Simplifying notation first makes the proof much easier to follow.
partition (verb): To split a set into separate, non-overlapping pieces that together cover it.
The lecture partitions A and B into disjoint regions C, D, and E.
manageable (adjective): Easy enough to handle or work with.
Naming the regions makes the algebra manageable.
insert (verb): To add something into an expression or equation.
Inserting plus P(E) minus P(E) is a common algebra trick.
derive (verb): To work out a result step by step from known rules.
The final formula is derived using only the additivity axiom.
event (noun): A subset of outcomes from a random experiment.
A and B are two events on the same sample space.
union (noun): The combination of two sets, containing everything in either one.
The union of C and D covers everyone in exactly one event.
intersection (noun): The outcomes that belong to two sets at the same time.
A intersect B is the region shared by both events.
complement (noun): Everything outside a given set within the whole sample space.
The complement of an event contains every outcome not in it.
identity (noun): A mathematical statement that is always true for any valid values.
The lecture proves a probability identity from the axioms.
axiom (noun): A basic rule accepted as a starting point for a theory.
Every step in the proof relies on the additivity axiom.
disjoint (adjective): Sharing no outcomes in common.
C, D, and E are disjoint regions of the diagram.
rigorous (adjective): Careful and exact, following clear logical steps.
The proof gives a rigorous derivation instead of relying on intuition.
expression (noun): A combination of symbols representing a mathematical quantity.
The messy expression is simplified using new notation.
term (noun): One part of a larger mathematical expression, added or subtracted.
Each term in the sum represents one disjoint region's probability.
algebraic manipulation (noun): The process of rearranging an equation using valid mathematical steps.
Algebraic manipulation turns the messy equation into a clean result.
step by step (phrase): Proceeding one small stage at a time, in order.
The proof builds the result step by step from the axioms.
label (verb): To give something a name so it can be referred to easily.
The lecture labels each disjoint region before working with it.
recitation problem (noun): A short practice question worked through in a smaller class session.
This is a short recitation problem following the axioms lecture.

From the YouTube description

MIT 6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013
View the complete course: http://ocw.mit.edu/6-041SCF13
Instructor: Kuang Xu

License: Creative Commons BY-NC-SA
More information at http://ocw.mit.edu/terms
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