Properties of a probability
Theorem taught in Random experiments and probability, part of Probability
Properties of a probability
| Type | Theorem |
| Taught in | Random experiments and probability |
| Part | Probability |
| Worked examples | 1 |
For events A and B,
- \mathbb{P}(\emptyset)=0.
- A\subset B implies \mathbb{P}(A)\le \mathbb{P}(B).
- \mathbb{P}(A)\le 1.
- \mathbb{P}(A^c)=1-\mathbb{P}(A).
- \mathbb{P}(A\cup B)=\mathbb{P}(A)+\mathbb{P}(B)-\mathbb{P}(A\cap B).
The empty set is disjoint from \Omega, so the sum rule gives \mathbb{P}(\Omega)=\mathbb{P}(\Omega)+\mathbb{P}(\emptyset). The complement formula is the sum rule on the disjoint union \Omega=A\cup A^c. For the union formula, write A\cup B as the disjoint union of A and B\cap A^c, and write B as the disjoint union of A\cap B and B\cap A^c. Subtracting the two expansions cancels \mathbb{P}(B\cap A^c).
Examples
A lower bound on an intersection
Question page: A lower bound on an intersection
Let \mathbb{P}(A)=0.9 and \mathbb{P}(B)=0.8. Show that \mathbb{P}(A\cap B)\ge 0.7.
The union formula rearranges to \mathbb{P}(A\cap B)=\mathbb{P}(A)+\mathbb{P}(B)-\mathbb{P}(A\cup B). The union has probability at most 1, so
\mathbb{P}(A\cap B)\ge 0.9+0.8-1=0.7.
Used in
- Taught in Random experiments and probability, where the derivation and examples are.
- All 17 questions on Random experiments and probability
See also
Sample space · Event · Probability measure · Equally likely outcomes · Four ways of counting draws · Conditional probability · Product rule · Law of total probability