Probability 15

The probability that the letter is in the table is p.
The probability that the letter is elsewhere is q.
So p = 1 - q and q = 1 - p
Chance in : chance not = p : q
Now seven drawers have been examined, so the chance that the letter is in the table is reduced to p/8.
Chance in : chance not = p/8 : q
Probability that the letter is in the last drawer = (p/8) / [(p/8) + q]
=p/(p + 8q)
=p/[p + 8(1-p)]
=p/(8 - 7p)
Or expressed in terms of q = (1 - q)/(1 + 7q)

This is a typical discrete Bayesian probability.
Pr (A|B) = Pr(A|C)*Pr(C|B)+Pr(A|C’)*Pr(C’|B), where Pr(C)+Pr(C’)=1

X = 1 if there is a letter in the writing table
X = 0 if there is no letter in the writing table
Pr (X = 1) = p
Pr (X = 0) = 1 – p

Y = 1 – 8 if there is a letter in the 1st – 8th drawer, given that X = 1
Pr (Y | X = 1) = 1 / 8 for any Y
Pr (Y | X = 0) = 0 for any Y
Pr (Y = 8 | X = 1) = Pr (Y not = 1 – 7 | X = 1) = 1 / 8

Pr (Y = 8 | Y not = 1 - 7)
= Pr (Y = 8 | X = 0) * Pr (X = 0 | Y not = 1 – 7) + Pr (Y = 8 | X = 1) * Pr (X = 1 | Y not = 1 – 7)
= Pr (Y = 8 | X = 0) * Pr (Y not = 1 – 7 | X = 0) * Pr (X = 0) / [Pr (Y not = 1 – 7 | X = 0) * Pr (X = 0) + Pr (Y not = 1 – 7 | X = 1) * Pr (X = 1)] + Pr (Y = 8 | X = 1) * Pr (Y not = 1 – 7 | X = 1) * Pr (X = 1) / [Pr (Y not = 1 – 7 | X = 0) * Pr (X = 0) + Pr (Y not = 1 – 7 | X = 1) * Pr (X = 1)]
= [Pr (Y = 8 | X = 0) * Pr (Y not = 1 – 7 | X = 0) * Pr (X = 0) + Pr (Y = 8 | X = 1) * Pr (Y not = 1 – 7 | X = 1) * Pr (X = 1)] / [Pr (Y not = 1 – 7 | X = 0) * Pr (X = 0) + Pr (Y not = 1 – 7 | X = 1) * Pr (X = 1)]

Alternatively, we assume the follows:

Y = 0 if there is no letter in the writing table, Pr (Y = 0) = 1 - p
Y = 1 if there is a letter in 1st drawer, Pr (Y = 1) = p / 8
Y = 2 if there is a letter in 2nd drawer, Pr (Y = 2) = p / 8
Y = 3 if there is a letter in 3rd drawer, Pr (Y = 3) = p / 8
Y = 4 if there is a letter in 4th drawer, Pr (Y = 4) = p / 8
Y = 5 if there is a letter in 5th drawer, Pr (Y = 5) = p / 8
Y = 6 if there is a letter in 6th drawer, Pr (Y = 6) = p / 8
Y = 7 if there is a letter in 7th drawer, Pr (Y = 7) = p / 8
Y = 8 if there is a letter in 8th drawer, Pr (Y = 8) = p / 8

Pr (Y = 0) + Pr (Y = 1) + Pr (Y = 2) + Pr (Y = 3 + Pr (Y = 4) + Pr (Y = 5) + Pr (Y = 6) + Pr (Y = 7) + Pr (Y = 8) = (1 – p) + (p / 8) + (p / 8) + (p / 8) + (p / 8) + (p / 8) + (p / 8) + (p / 8) + (p / 8) = 1

2009-07-09 09:52:05 補充：
Pr (Y = 8 | Y not = 1 – 7) = Pr (Y = 8) / {1 – [Pr (Y = 1) + Pr (Y = 2) + Pr (Y = 3 + Pr (Y = 4) + Pr (Y = 5) + Pr (Y = 6) + Pr (Y = 7)]} = (p/ 8) / {1 – [p / 8) + (p / 8) + (p / 8) + (p / 8) + (p / 8) + (p / 8) + (p / 8)]} = (p / 8) / (1 – 7 * p / 8) = p / (8 – 7 * p)

2009-07-09 09:54:51 補充：
Pr (Y = 8 | Y not = 1 – 7) = Pr (Y = 8) / {1 – [Pr (Y = 1) + Pr (Y = 2) + Pr (Y = 3 + Pr (Y = 4) + Pr (Y = 5) + Pr (Y = 6) + Pr (Y = 7)]} = (p/ 8) / {1 – [p / 8) + (p / 8) + (p / 8) + (p / 8) + (p / 8) + (p / 8) + (p / 8)]} = (p / 8) / (1 – 7 * p / 8) = p / (8 – 7 * p)

2009-07-09 09:55:20 補充：
Pr (Y = 8 | Y not = 1 – 7) = Pr (Y = 8) / {1 – [Pr (Y = 1) + Pr (Y = 2) + Pr (Y = 3 + Pr (Y = 4) + Pr (Y = 5) + Pr (Y = 6) + Pr (Y = 7)]} = (p/ 8) / {1 – [p / 8) + (p / 8) + (p / 8) + (p / 8) + (p / 8) + (p / 8) + (p / 8)]} = (p / 8) / (1 – 7 * p / 8) = p / (8 – 7 * p)

2009-07-09 09:55:30 補充：
Pr (Y = 8 | Y not = 1 – 7) = Pr (Y = 8) / {1 – [Pr (Y = 1) + Pr (Y = 2) + Pr (Y = 3 + Pr (Y = 4) + Pr (Y = 5) + Pr (Y = 6) + Pr (Y = 7)]} = (p/ 8) / {1 – [p / 8) + (p / 8) + (p / 8) + (p / 8) + (p / 8) + (p / 8) + (p / 8)]} = (p / 8) / (1 – 7 * p / 8) = p / (8 – 7 * p)

2009-07-09 10:16:22 補充：
Pr (Y = 8 | Y not = 1 – 7) = Pr (Y = 8) / {1 – [Pr (Y = 1) + Pr (Y = 2) + Pr (Y = 3 + Pr (Y = 4) + Pr (Y = 5) + Pr (Y = 6) + Pr (Y = 7)]} = (p/ 8) / {1 – [p / 8) + (p / 8) + (p / 8) + (p / 8) + (p / 8) + (p / 8) + (p / 8)]} = (p / 8) / (1 – 7 * p / 8) = p / (8 – 7 * p)

2009-07-09 10:16:31 補充：
OK?