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o = . The odds of an event E given an hypothesis H are written as O(E|H). Sometimes odds are expressed as a ratio such as '3:2' or '3 to 2'. This means o=3/2. The following phrases are equivalent.
'a:b', 'a:b on', 'o = a/b', 'a to b', 'b to a against' etc.
(c) Special values of p.
'Certainty' 'Impossibility' 'Evens' |
p=1 or o=∞ p=0 or o=0 p=½ or o=1. |
(d) Relationship of Events.
Two events are 'mutually exclusive' if they cannot both happen. Two events are 'independent' if a knowledge that one is true does not affect the probability of the other one. A number of events is 'exhaustive' if it is certain that one or other of them will happen.
(e) The laws of probability.
(i) the law of addition of probability | |
P(E_{1}∨E_{2}|H) = P(E_{1}|H) + P(E_{2}|H) | |
if E1 and E2 are mutually exclusive. | |
(ii) the law of multiplication of probabilities | |
P(E_{1}E_{2}|H) = P(E_{1}|H) P(E_{2}|E_{1}H). | |
In particular, if E1 and E2 are independent | |
P(E_{1}E_{2}|H) = P(E_{1}|H) P(E_{2}|H) |
(f) Some theorems.
(i) P(E_{1}E_{2} ... E_{n}|H) | |
= P(E_{1}|H) P(E_{2}|E_{1}H) P(E_{3}|E_{1}E_{2}H) ... P(E_{n}|(E_{1} ... E_{n-1}H). | |
(ii) P(E_{1}∨E_{2}∨ ... ∨E_{n}|H) | |
- ... etc., |
and in particular if E_{1}, E_{2}, ..., E_{n} are all mutually exclusive, the right hand side can be replaced by . If E_{1}∨ ... ∨E_{r} is exhaustive the left hand side is 1. Therefore .
(iii) Bayes' theorem. | |
For various hypotheses H_{i} (i = 1, 2...) | |
The proof of this is simple. For by the law of multiplication of probabilities.
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