Book Probability by Jim Pitman, Page 19
Events and Sets
Set Theory & Venn Diagram Gallery
Intersection: \(A \cap B\)
“A AND B” — Elements in both.
Union: \(A \cup B\)
“A OR B” — Elements in either or both.
Complement: \(A^c\)
“NOT A” — Everything outside A.
Subset: \( A \subseteq B \)
A is entirely contained within B.
Difference: \( A \setminus B \)
Elements in A but NOT in B.
Disjoint: \( A \cap B = \emptyset \)
No overlap between sets.
Universal Set: \( U \)
The set of all possible elements.
Empty Set: \( \emptyset \)
A set with zero elements.
Partitions
Set Partition Visualization
A collection of subsets is a partition if:
- Non-Empty: None of the sets are empty.
- Collectively Exhaustive: The union of all sets equals the original set S.
- Mutually Exclusive: No two sets overlap (Intersection is empty).
Rules of Proportion and Prbability
Rules of Probability & Proportion
1. Complement Rule
\( P(A^c) = 1 – P(A) \)
The probability of an event NOT happening is 1 minus the probability that it does.
2. Difference Rule
\( P(A \setminus B) = P(A) – P(A \cap B) \)
To find the probability of “A but not B,” subtract the overlap from A.
3. Inclusion-Exclusion Rule
\( P(A \cup B) = P(A) + P(B) – P(A \cap B) \)
The “General Addition Rule.” We subtract the intersection so it isn’t counted twice.
The Rule of Total Proportion
For any sample space \( S \), the sum of probabilities for all mutually exclusive and exhaustive events must equal 1. \[ \sum P(E_i) = 1 \]
Distribution
In probability and statistics, a distribution is a mathematical function that describes all the possible values a random variable can take and how frequently those values occur.
Think of it as a “map” that tells you where the probability is concentrated. If you were to roll a die, the distribution would tell you that the numbers 1 through 6 each have an equal probability of 1/6.
Common Named Distributions
Normal
Symmetrical “Bell Curve” (Height, Test Scores)
Uniform
Equal probability (Rolling a die, Random.org)
Exponential
Rapid decay (Time until next phone call)