Brother and Sister

Originally posted by wolfgang59
WRONG

If that were true it would mean that every 2 child family with a daughter had a 2/3 chance of having a son and daughter.

Switch genders and we can then say every 2 child family with a son has a 2/3 chance of having a son and a daughter.

So therefore EVERY 2 child faamhaving a son and daughter - which is clearly false.

I cannot understand your final conclusion (the one you state is clearly false).

Regardless, your anlaysis here is flawed. Thomaster had basically asserted (before later retracting it) that P[family F has brother & sister | family F has a daughter named Mary] = 2/3. Regardless of whether or not he was right, your analysis assumes that P[F has brother & sister | F has a daughter named Mary] equates to P[F has brother & sister | F has at least one daughter]. But that turns out to be false, just like it is generally false that P[X | I1] equates to P[X | I2] when information I1 happens to entail I2 but I2 does not entail I1.

Originally posted by wolfgang59
WRONG

If that were true it would mean that every 2 child family with a daughter had a 2/3 chance of having a son and daughter.

Switch genders and we can then say every 2 child family with a son has a 2/3 chance of having a son and a daughter.

So therefore EVERY 2 child faamhaving a son and daughter - which is clearly false.

Originally posted in much haste!

WRONG

If that were true it would mean that every 2 child family with a daughter had a 2/3 chance of having a son and daughter.

Switch genders and we can then say every 2 child family with a son has a 2/3 chance of having a son and a daughter.

So therefore EVERY 2 child family having either a son or daughter has a 2/3 chance of having a son and a daughter- which is clearly false.

We debated this about 18 months ago (but I cannot find it). I first posed the question as a coin toss problem.

Originally posted by wolfgang59
Originally posted in much haste!

WRONG

If that were true it would mean that every 2 child family with a daughter had a 2/3 chance of having a son and daughter.

Switch genders and we can then say every 2 child family with a son has a 2/3 chance of having a son and a daughter.

So therefore EVERY 2 child family having either a son or daughter ha ...[text shortened]... s about 18 months ago (but I cannot find it). I first posed the question as a coin toss problem.

In that case, I have another objection against your analysis in addition to my previous one. Your analysis seems to rest on the following assumption (call it A), which I think is generally false:

A: If P[X | I1] = m; and if P[X | I2] = m; then P[X | I1 OR I2] = m.

(Here in this case, X = F has a son & a daughter; and I1 = F has at least one daughter; and I2 = F has at least one son, and m=2/3).

Am I correct that your analysis relies on A? If so, do you have some justification for using this? Because as far as I can tell, it is generally false and only holds for certain cases. Consider:

P[X | I1 OR I2] = P[X & (I1 OR I2)]/P[I1 OR I2]
... = P[(X & I1) OR (X & I2)]/P[I1 OR I2]
... = {P[X & I1] + P[X & I2] - P[(X & I1) & (X & I2)]}/P[I1 OR I2]
... = {P[X & I1] + P[X & I2] - P[X & (I1 & I2)]}/{P[I1] + P[I2] - P[I1 & I2]}

By assumption here P[X & I1]/P[I1] = P[X & I2]/P[I2] = m. So it should follow that P[X | I1 OR I2] = m if and only if the following is satisfied:

P[X & (I1 & I2)] = m*P[I1 & I2].

But as far as I can tell, this only holds for the following cases:
(a) It holds if I1 and I2 are mutually exclusive.
(b) Else, it holds if X and [I1 & I2] are independent and P[X] =m; or if P[X | (I1 & I2)] = m (= P[X | (I1 OR I2)]).

But since none of these should hold in this case, your reductio doesn't work, since one should simply reject A, rather than the premises that P[X | I1] = P[X | I2] = m.

What am I missing?