Probability theory: Difference between revisions
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The basic theorems of probability can be developed easily from the [[ |
The basic theorems of probability can be developed easily from the [[probability axioms]] and [[Set Theory]]. |
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Pr[A<sub>1</sub> * A<sub>2</sub>] = |
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#<font size=+1 color=red> Pr[A<sub>1</sub> * A<sub>2</sub>] =</font> <font size=+2 color=red>Σ</font><font size=+1 color=red><sub>E''i''</sub> ε A<sub>1</sub> &inter; A<sub>2</sub> Pr[E<sub>i</sub>] for all E<sub>i</sub> in both A<sub>1</sub> and A<sub>2</sub>.</font> |
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<font size=+2>Σ</font> Pr[E<sub>i</sub>] |
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Where E<sub>''i''</sub> is any event in both A<sub>1</sub> and A<sub>2</sub>. |
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Pr[A<sub>1</sub> + A<sub>2</sub>] = |
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<font size=+2>Σ</font> Pr[E<sub>i</sub>] |
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Where E<sub>''i''</sub> is any event in either A<sub>1</sub> or A<sub>2</sub>. |
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#<font size=+1 color=red> Pr[A<sub>1</sub> + A<sub>2</sub>] = Pr[A<sub>1</sub>] + Pr[A<sub>2</sub>] - Pr[A<sub>1</sub> * A<sub>2</sub>]</font> |
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The probability of some event happening knowing that another event happened before can be computed using [[ |
The probability of some event happening knowing that another event happened before can be computed using [[conditional probability]]. |
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Revision as of 06:29, 3 July 2001
The basic theorems of probability can be developed easily from the probability axioms and Set Theory.
- The sum of the probabilities of all the elementary events is one.
- For any arbitrary events A1 and A2, the probability of both events is given by the sum of the probabilities for all elementary events in both A1 and A2. If the intersection is empty, then the probability is exactly zero.
- For any arbitrary events A1 and A2, the probability of either or both is given by the sum of the probabilities of the two events minus the probability of both.
The formulae below express the same ideas in algebraic terms.
| Σi Pr[Ei] = 1 |
Pr[A1 * A2] =
Σ Pr[Ei]
Where Ei is any event in both A1 and A2.
Pr[A1 + A2] =
Σ Pr[Ei]
Where Ei is any event in either A1 or A2.
(in these equations, "+" means "or" and "*" means "and")
The probability of some event happening knowing that another event happened before can be computed using conditional probability.