Event
Definition taught in Random experiments and probability, part of Probability
Event
| Type | Definition |
| Taught in | Random experiments and probability |
| Part | Probability |
| Properties | 1 |
| Worked examples | 1 |
An event is a subset of the sample space. The event occurs when the outcome lies in that subset.
Properties
Properties of a probability
Main article: Properties of a probability
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 four-engine aeroplane
Worked example from Random experiments and probability
An aeroplane has engines 1 and 2 on the left wing and 3 and 4 on the right. It can fly as long as at least one engine on each wing works. Each engine fails during a flight with probability q, independently of the others.
Question. Describe the sample space and the event that the aeroplane can fly, and find its probability.
Solution. Record which engines work: an outcome is a vector in \{0,1\}^4, so there are 2^4=16 outcomes. Let A_i be the event that engine i works. The aeroplane can fly on the event
(A_1\cup A_2)\cap(A_3\cup A_4).
A wing fails only if both its engines fail, with probability q^2, so by independence of the two wings
\mathbb{P}\{(A_1\cup A_2)\cap(A_3\cup A_4)\}=(1-q^2)^2.
With q=0.1 that is 0.9801. A single engine fails one flight in ten, but the aeroplane only about one flight in fifty. Redundancy turns unreliable parts into a reliable system, which is the subject of Reliability.
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 · Probability measure · Properties of a probability · Equally likely outcomes · Four ways of counting draws · Conditional probability · Product rule · Law of total probability