Return to the list of my pages written in English about the two envelopes problem
X(ωr,L) = r.
X(ωr,G) = 2r.
Y(ωr,L) = 2r.
Y(ωr,G) = r.
If amount of money is real number and we have illusion of probability we can make a mistake in the expectation formula and we can make a paradox.
X(ωn,L) = n.
X(ωn,G) = 2n.
Y(ωn,L) = 2n.
Y(ωn,G) = n.
If amount of money is natural number the expectation formula is not well formed and there can be no paradox.
Return to the list of my pages written in English about the two envelopes problem
2015/03/22 12:35:25
First edition 2015/02/13
If amout of money is natural number the two envelopes problem is not established.
Caution
I who am Japanese wrote this page in English, but I am not so good at English.
I who am Japanese wrote this page in English, but I am not so good at English.
If amount of money is real number
Elementary events
Let r be a positive real number.
Let ωr, L denote the event that the lesser amount is r and chosen amount is lesser amount.
Let ωr, G denote the event that the lesser amount is r and chosen amount is greater amount.
Let ωr, L denote the event that the lesser amount is r and chosen amount is lesser amount.
Let ωr, G denote the event that the lesser amount is r and chosen amount is greater amount.
Sample space
Ω = { ωr,K | r is a real number ∧ K is L or G }
Probability space
Let (Ω, F, P) be a probablity space.
Random variables
Let X, Y be randome variables for the amount of money in the chosen envelope and another envelope respectively.X(ωr,G) = 2r.
Y(ωr,L) = 2r.
Y(ωr,G) = r.
Conditional expectation
Let fL(x) be probability density function of ωx,L .
Let fG(x) be probability density function of ωx,G .
If x is a arbitrary positive real number
E[Y|X=x]
= E[Y|ωx,L ∪ ωx/2,G]
= E[Y|ωx,L]fL(x)/(fL(x)+fG(x/2))
+ E[Y|ωx/2,G]fG,(x/2)/(fL(x)+fG(x/2))
= 2xfL(x)/(fL(x)+fG(x/2)) + (x/2)fG,(x/2)/(fL(x)+fG(x/2))
< < < illusion of probability > > >
= 2x × (1/2) + (x/2) × (1/2)
< < < paradox > > >
= 1.25 x.
Let fG(x) be probability density function of ωx,G .
If x is a arbitrary positive real number
E[Y|X=x]
< < < illusion of probability > > >
< < < paradox > > >
= 1.25 x.
If amount of money is real number and we have illusion of probability we can make a mistake in the expectation formula and we can make a paradox.
If amount of money is natural number
Elementary events
Let n be a natural number.
Let ωn, L denote the event that the lesser amount is n and chosen amount is lesser amount.
Let ωn, G denote the event that the lesser amount is n and chosen amount is greater amount.
Let ωn, L denote the event that the lesser amount is n and chosen amount is lesser amount.
Let ωn, G denote the event that the lesser amount is n and chosen amount is greater amount.
Sample space
Ω = { ωn,K | n is a natural number ∧ K is L or G }
Probability space
Let (Ω, F, P) be a probablity space.
Random variables
Let X, Y be randome variables for the amount of money in the chosen envelope and another envelope respectively.X(ωn,G) = 2n.
Y(ωn,L) = 2n.
Y(ωn,G) = n.
Conditional expectation
If x is a arbitrary natural number
E[Y|X=x]
= E[Y|ωx,L ∪ ωx/2,G]
= E[Y|ωx,L]P(ωx,L)/P(ωx,L ∪ ωx/2,G)
+ E[Y|ωx/2,G]P(ωx/2,G)/P(ωx,L ∪ ωx/2,G).
< < < ω(x/2),G may be empty event, and value of Y is indeterminate. > > >
E[Y|X=x]
< < < ω(x/2),G may be empty event, and value of Y is indeterminate. > > >
If amount of money is natural number the expectation formula is not well formed and there can be no paradox.
Return to the list of my pages written in English about the two envelopes problem