jt,
- Please show me where you think that this has been demonstrated algebraically -- I don't think that it has.
Actually, the proof's quite simple.
(1) P(H) + P(~H)=1, by definition.
(2) P(E|H)P(H) + P(E|~H)P(~H) = P(E), from Bayes' Theorem.
(3) P(E) = 1, as stipulated by Jabba.
Suppose P(E|H) < 1. Then,
P(E|H)P(H) < P(H) (multiplying both sides by P(H))
P(E|H)P(H) + P(~H) < P(H) + P(~H) (adding P(~H) to both sides)
P(E|H)P(H) + P(~H) < 1 (substituting from (1))
P(E|H)P(H) + P(~H) < P(E) (substituting from (3))
P(E|H)P(H) + P(~H) < P(E|H)(P(H) + P(E|~H)P(~H) (substituting from (1))
P(~H) < P(E|~H)P(~H) (subtracting P(E|H)P(H) from both sides)
1 < P(E|~H) (dividing both sides by P(~H))
But this implies a probability greater than 1, which is impossible. Therefore P(E|H) cannot be less than one.
As it is a probability, 0 <+ P(E|H) <= 1.
Therefore, P(E|H) = 1.
Now, we know from (1) and (2) that P(E|H)P(H) + P(E|~H)P(~H) = 1.
P(H) + P(E|~H)P(~H) = 1 (substituting for P(E|H))
P(H) + P(E|~H)P(~H) = P(H) + P(~H) (substituting from (1))
P(E|~H)P(~H) = P(~H) (subtracting P(H) from both sides)
P(E|~H) = 1 (dividing both sides by p(~H))
Therefore, P(E|H) = P(E|~H) = 1
QED.
Dave