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The field of probability is defined by axioms. If S is a set of numbers containing certain members then:
| Axiom 1: | if A is an event in S, then P(A) >= 0. |
| Axiom 2: | P(S) = 1. The probability of the whole sample space is one. |
| Axiom 3: | if A and B are mutually exclusive then P ( A ∪ B ) = P(A) + P(B). In other words, if the separate events of A and B are counted together, their probabilities are added. |
As a result of these axioms, probability can be defined by certain properties.
| 1. | P(Ac) = 1 - P(A) | The probability of an event not occurring is the same as one minus the probability of that event occurring. |
| 2. | P(∅) = 0 | The probability of no event happening is 0. |
| 3. | If A ⊆ B then P(A) <= P(B) | If that event is part of a larger event, then it's probability is less. |
| 4. | P(A) <= 1 | The probability of any event is less than or equal to 1. |
| 5. | P(A ∪ B) = P(A) + P(B) - P(A ∩ B) | If events A and B are joined together, then their probabilities are added together minus where they overlap. |
Let the probability of S or P(S) be defined by the area of the rectangle of S and to be equal to 1. Then P(A) is less than 1 and P(B) is less than 1. The union of A and B, P(A ∪ B) includes an area that overlaps. Since this overlap can only be counted once, its area of intersection is subtracted out. This area of intersection is P(A ∩ B).
If the sample space has a finite number of outcomes it is discrete. If a is an element of A, the discrete probability function would be:
P(A) = ∑a ∈ A P(a)
If the sample space has an infinite number of outcomes it is continuous. If A is an event in S and y is the outcome, then:
P(A) = ∫Af(y)dy
The area defined by A of the function of y is the probability of event A.
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