stats : bayes theorem Watch

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kikzen
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#1
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#1
Your wallet contains a single note which you think is equally likely to
be either £10 or £20. You add a £10 note without looking. Later, you
remove one of the two notes at random, again without looking inside
the wallet. It's a £10 note. What is the chance the other note is £20?
try first then see what i did wrong (in white)

i said

let A be event £10, B be event £20
P(A) = P(B) = 1/2

(B^c is complement of B)

then P(B|A) = P(A|B)P(B) / [P(A|B)P(B) + P(A|B^c)P(B^c)] by Bayes Theorem

= 0.5*0.5 / (0.5*0.5 + 0.5*0.5)
=1/2

but the answer is 1/3 ?!
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Gauss
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#2
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#2
(Original post by kikzen)
Your wallet contains a single note which you think is equally likely to
be either £10 or £20. You add a £10 note without looking. Later, you
remove one of the two notes at random, again without looking inside
the wallet. It's a £10 note. What is the chance the other note is £20?
try first then see what i did wrong (in white)

i said

let A be event £10, B be event £20
P(A) = P(B) = 1/2

(B^c is complement of B)

then P(B|A) = P(A|B)P(B) / [P(A|B)P(B) + P(A|B^c)P(B^c)] by Bayes Theorem

= 0.5*0.5 / (0.5*0.5 + 0.5*0.5)
=1/2

but the answer is 1/3 ?!

P(B^c) = 1, not (1/2)

P(B|A) = P(A|B)P(B) / [P(A|B)P(B) + P(A|B^c)P(B^c)]

=> 0.5*0.5 / (0.5*0.5 + 0.5*1)
=> (1/4)/(1/4 + 1/2)
=> (1/4)/(3/4)
=> 1/3


Euclid.
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[email protected]
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#3
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#3
Call the total of the notes x so to start with
p(x=20)=0.5
p(x=10)=0.5
then you add a £10 note
=> p(x=30)=0.5 p(x=20)=0.5
you take out a note, y
p(y=10)=0.5 (it being the 2nd note) + 0.5x0.5 (it being the first note, which is a £10 note)= *0.75* =>p(y=20)=*0.25*
Turn out its a £10 note. (ie probability=0.75)
So *given that* its a £10, whats the chance the other is a £20:
0.25/0.75 = 1/3
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kikzen
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#4
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is P(B^c) = 1 because you already have £10 in your hand? if not ,why not?!
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Gauss
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#5
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(Original post by kikzen)
is P(B^c) = 1 because you already have £10 in your hand? if not ,why not?!
Yes, P(B^c) is the chance of already having chosen £10, which is 1.

Euclid.
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kikzen
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#6
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euclid youll get some rep soon; gotta spread it around first apparently
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Gauss
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#7
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(Original post by kikzen)
euclid youll get some rep soon; gotta spread it around first apparently
thank you.
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[email protected]
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thank you kikzen for my rep!
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kikzen
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#9
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ya welcome
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