Events, Probability Laws, and Complements

Scope Label

Core 9758. This note covers the language and laws needed before conditional probability and independence.

Use it with the hub Probability.

Outcomes, Sample Spaces, and Events

An experiment is a chance process.

An outcome is one possible result.

The sample space $S$ is the set of all possible outcomes.

An event is a subset of $S$.

For a fair die,

$$S=\{1,2,3,4,5,6\}.$$

If

$$A=\{\text{even}\}=\{2,4,6\},B=\{\text{prime}\}=\{2,3,5\},$$

then $A$ and $B$ are events inside the same sample space.

Caption: In a die experiment, events such as “even” and “prime” are subsets of the same sample space.

Why the Sample Space Matters

The sample space is not automatically every possible thing that could happen. It is the set of outcomes for the experiment currently being considered.

Changing the experiment can change the sample space. For example:

• throwing one die has sample space $\{1,2,3,4,5,6\}$
• throwing two dice has ordered-pair outcomes such as $(1,1),(1,2),\dots ,(6,6)$
• choosing a card has outcomes based on cards, not die values

If the sample space is wrong, the probability calculation is usually wrong even if the formula is applied correctly.

Caption: Probability starts by identifying the full sample space and then locating events as subsets inside it.

Event Operations

Union

$A\cup B$ means $A$ or $B$ or both.

The word “or” is usually inclusive in probability unless the question explicitly says otherwise.

Intersection

$A\cap B$ means both $A$ and $B$ occur.

This is the overlap of the two events.

Complement

$A^c$ means not $A$.

The complement includes all outcomes in $S$ that are not in $A$.

Mutually exclusive events

Events $A$ and $B$ are mutually exclusive if they cannot occur together:

$$A\cap B=\varnothing .$$

In probability notation,

$$P(A\cap B)=0.$$

Caption: Union, intersection, and complement describe how events occupy regions within the same sample space.

Translating Words into Notation

Many probability errors begin with mistranslation.

Words	Notation
$A$ or $B$	$A\cup B$
$A$ and $B$	$A\cap B$
not $A$	$A^c$
$A$ but not $B$	$A\cap B^c$
exactly one of $A$ and $B$	$(A\cap B^c)\cup (A^c\cap B)$
at least one of $A$ and $B$	$A\cup B$
neither $A$ nor $B$	$(A\cup B{)}^c$

Caption: Probability questions often begin by translating ordinary language into event notation.

What Probability Means

A probability is a number between $0$ and $1$:

$$0\le P(A)\le 1.$$

If all outcomes in a finite sample space are equally likely, then

$$P(A)=\frac{\text{number of outcomes in }A}{\text{number of outcomes in }S}.$$

This counting ratio should only be used directly when the outcomes are equally likely.

Caption: Counting methods apply directly only when the outcomes in the sample space are equally likely.

Basic Laws of Probability

Complement law

Since $A$ and $A^c$ together make the whole sample space,

$$P(A^c)=1-P(A).$$

General addition law

For any two events,

$$P(A\cup B)=P(A)+P(B)-P(A\cap B).$$

The overlap $A\cap B$ is subtracted because it is counted once in $P(A)$ and once again in $P(B)$.

Caption: The addition law subtracts the overlap because it is counted twice in $P(A)+P(B)$.

Mutually exclusive addition law

If $A$ and $B$ are mutually exclusive, then $P(A\cap B)=0$, so

$$P(A\cup B)=P(A)+P(B).$$

This is a special case, not the general rule.

Complement Thinking

Complement thinking is useful when the direct event has many cases but the opposite event is simple.

The general idea is:

$$P(\text{wanted})=1-P(\text{unwanted}).$$

At least one

“At least one” often means:

$$P(\text{at least one})=1-P(\text{none}).$$

For example, if three independent components each work with probability $0.9$, then

$$P(\text{at least one fails})=1-P(\text{none fail})=1-0.9^3.$$

Caption: “At least one” is often easier to solve by subtracting the case where none occur.

Common complement patterns

Direct phrase	Complement
at least one success	no successes
not all work	all work
at most $k$	more than $k$
no more than $k$	at least $k+1$
at least $k$	fewer than $k$

Worked Core Examples

Example 1: Addition law

Suppose

$$P(A)=0.6,P(B)=0.5,P(A\cap B)=\mathrm{0.2.}$$

Then

$$P(A\cup B)=0.6+0.5-0.2=\mathrm{0.9.}$$

Example 2: Mutually exclusive events

If $A$ and $B$ are mutually exclusive and

$$P(A)=0.3,P(B)=0.4,$$

then

$$P(A\cup B)=0.3+0.4=\mathrm{0.7.}$$

We do not subtract an overlap because there is no overlap.

Example 3: Complement

If a fair die is thrown twice, find the probability of at least one six.

The complement is no six on either throw:

$$P(\text{at least one six})=1-{\left(\frac{5}{6}\right)}^2=\frac{11}{36}.$$

Common Pitfalls

• Treating “or” as exclusive when the question does not say so.
• Using $P(A\cup B)=P(A)+P(B)$ without checking overlap.
• Forgetting that $A^c$ depends on the current sample space.
• Using counting ratios when outcomes are not equally likely.
• Counting “at least one” directly when the complement is simpler.

Links

• Probability
• Conditional Probability, Independence, and Trees
• Permutations and Combinations