- Describe sample spaces and events with set notation.
- Apply the probability axioms and their main consequences.
- Calculate probabilities by counting equally likely outcomes.
- Distinguish permutations, combinations, independence, and mutual exclusivity.
- Use conditional probability, total probability, and Bayes’ theorem.
- Basic arithmetic and algebra.
- Familiarity with elementary set notation is helpful.
Outcomes, sample spaces, and events
Probability begins by organizing experimental outcomes as sets.
Probability uses sets to describe the possible results of an experiment and the groups of results whose likelihoods matter.
- One die: S={1,2,3,4,5,6}.
- One coin toss: S={heads,tails}.
- Two dice: each result is ordered, such as (1,3) or (6,2).
- A={2}: the die shows 2.
- B={2,4,6}: the result is even.
- C={1,2,3,4}: the result is below 5.
- S: the certain event containing every outcome.
- ∅: the event with no outcomes.
How events relate as sets
Containment, union, intersection, and complement express logical connections among events.
- A⊆B implies A∪B=B.
- If A⊆B and B⊆A, then A=B.
- A⊆B implies A∩B=A.
- A∪Aᶜ=S.
The associative, commutative, and distributive laws from set theory remain valid for probability events.
Exclusive events, exhaustive events, and partitions
These ideas describe whether events overlap and whether they cover all possibilities.
On one die, “odd” and “even” are disjoint because no result has both properties.
For one die, classify “result below 4” and “result above 3” as mutually exclusive, exhaustive, both, or neither.
Both. The sets {1,2,3} and {4,5,6} do not overlap, and together they equal S.
The three probability axioms
A probability assigns numbers to events under three required conditions.
- Nonnegativity: ≥0 for every event A.
- Normalization: =1.
- Countable additivity: for disjoint A₁,A₂,…, =∑ᵢ.
The last rule covers finite and infinite disjoint sequences. Any function meeting all three axioms is a probability, and one sample space may support different such functions. Texts may call the object a probability function or probability distribution.
Consequences of the axioms
Set relationships and the axioms yield several standard probability rules.
- Every event satisfies .
- =1−, since A and Aᶜ are disjoint and exhaustive.
- =0.
- If A⊆B, then ≤.
- =+−.
- For disjoint A and B, =+.
- =−.
The complement rule can replace a difficult event with its easier opposite—for instance, finding “at least one success” through “no successes.” In the addition rule, the intersection is removed because adding and counts shared outcomes twice.
Why does the addition rule subtract ?
The common outcomes already appear once in and once in , so subtraction corrects the double count.
Equally likely finite outcomes
In a simple sample space, probability reduces to counting.
This ratio obeys the axioms: counts are nonnegative; /=1; and disjoint sets satisfy . The practical work is therefore counting all outcomes and the favorable ones.
License plates with and without reuse
Whether a selected character remains available changes the counting model.
What creates the difference between the two plate models?
Without replacement, each chosen character reduces the number available for the next position; with replacement, the number stays at 36.
Counting rules and combinatorics
Sequential choices, ordered arrangements, and unordered selections require different rules.
- Multiplication rule: stages with m and n choices produce m·n outcomes. More stages are handled by multiplying every stage count; 4·3·5=60.
- Permutation: an ordered arrangement. All n objects have n! orders, while choosing and ordering n from N gives N!/(N−n)!.
- Combination: an unordered selection. The count is =N choose n=N!/[n!(N−n)!]. Dividing by n! removes repeated orderings.
License plates use permutations because ABC123 and BCA231 are distinct arrangements.
Blocks and category selections
Two examples show how structure determines a favorable count.
The numerator counts selections with the required category totals, while the denominator counts all n-topping selections. This pattern underlies the hypergeometric distribution for category sampling without replacement.
Why are combinations used for the topping calculation?
Reversing the order of the same two toppings does not produce a different pizza selection.
Independence
Independent events do not provide probability-changing information about each other.
If A and B are independent, so are A and Bᶜ. For several independent events, the probability of their intersection equals the product of their individual probabilities.
Repeated independent trials
Multiplication handles a fixed sequence, while combinations count sequences sharing the same success total.
Assume a basketball player makes each three-point shot with probability 0.44, misses with probability 0.56, and takes shots independently with unchanged probabilities.
When a problem asks for “at least one,” which opposite event should you consider first?
Check whether “none” is simpler to calculate, then subtract its probability from 1.
Conditional probability
Conditioning restricts attention to outcomes compatible with observed information.
Knowing B occurred removes outcomes outside B. The relevant sample space becomes B, and its favorable portion for A is A∩B. The positive-probability condition prevents division by zero.
Thus, for >0, independence is equivalent to =: observing B supplies no probability-changing information about A.
The law of total probability
A partition lets an overall event be calculated case by case.
Suppose A₁,…,Aₖ form a partition. For any event W, distributing W across their union gives W=(W∩A₁)∪⋯∪(W∩Aₖ). These pieces remain mutually exclusive.
This law obtains a total probability by adding the contribution from every mutually exclusive case. With the two-case partition A and Aᶜ, =P(B|A)$$P(A)+P(B|Aᶜ)$$P(Aᶜ).
Reversing a condition with Bayes’ theorem
Bayes’ theorem converts information about B given A into a probability for A given B.
Conditional probability gives =P(B|A)$$P(A) and also =P(A|B)$$P(B). Equating the two expressions produces Bayes’ rule.
- is the prior probability.
- measures how likely the evidence is when A holds.
- includes every route by which the evidence may arise.
- is the probability after B has been observed.
Medical testing and a rare virus
A low base rate can keep the probability after a positive test surprisingly small.
The prior, or base rate, explains the result. Among 1,000 people, about one is infected and is detected with probability 0.99. About 999 are uninfected, and roughly 5% of that much larger group—around 50 people—receive false positives. Positive results therefore contain many more false positives than true positives.
A positive result is useful but does not by itself make a rare infection highly probable when the test is imperfect. Further evidence, such as symptoms or another test, can be incorporated sequentially or through a multi-case Bayes calculation when the needed conditional probabilities are available.
Common misconceptions
Watch out: Independent events must come from separate experiments.
Instead: Events defined from the same die roll can be independent when their intersection probability equals the product of their probabilities.
Watch out: Mutual exclusivity and independence describe the same relationship.
Instead: Exclusivity forbids joint occurrence; independence means observing one event does not alter the probability of the other.
Watch out: Changing the order of selected objects never creates another outcome.
Instead: Order distinguishes dice pairs and license plates, but it does not distinguish a handshake pair or an unordered topping selection.
Watch out: A positive result from a fairly reliable test makes a rare infection highly probable.
Instead: The prior rate and false positives must also be included; in the given example, the updated probability is only about 1.94%.
Lesson recap
- A sample space lists every outcome; events are its subsets. Union means “or,” intersection means “and,” and complement means “not.”
- Disjoint events do not overlap, exhaustive events cover S, and a collection with both properties is a partition.
- Probabilities are nonnegative, assign 1 to S, and add across disjoint events.
- For finite equally likely outcomes, =/.
- Multiply across stages; use permutations for ordered outcomes, combinations for unordered selections, and factorials to abbreviate products.
- Independence is characterized by =P(A)$$P(B).
- Independent sequence probabilities multiply, while combinations count sequences with a specified number of successes.
- Conditional probability uses =/ to restrict attention to B.
- Across a partition, =∑ᵢP(W|Aᵢ)$$P(Aᵢ).
- Bayes’ theorem reverses conditional direction and updates a prior probability with evidence while accounting for every way that evidence can occur.
Probability connects set logic, counting, independence, and evidence-based updating: counting handles equally likely cases, conditioning incorporates information, and Bayes’ theorem shows how that information changes an initial probability.