I don’t think you have the dependencies quite right, because you can actually use more of the information than you do above to restrict the population from which you draw.
The real underlying population you should draw on seems to to be the population of fathers with exactly two children, of which one might be a boy born on Tuesday.
p(a two boy family given one brother was born on Tuesday) = (p(one brother born on Tuesday in a in two-boy family)) (p(two boys in 2 person families)) / p(out of all two person families, having one be a boy born on Tuesday)
which is if we say Tuesday birth is 1⁄7 and boy is 1⁄2,
(2/7) (1/4) / (2/14) = 1⁄2 so the Tuesday datum drops out.
You’re not stating what probability rules (theorems/axioms) you are using (you’re probably going by intuition), and you have made mistakes.p(one brother born on Tuesday in a in two-boy family) is not 2⁄7; It’s 1/7 + 1/7 - (1/7)(1/7) because you’re counting the two children both being born on Tuesday twice. The same mistake has been made in calculating p(out of all two person families, having one be a boy born on Tuesday); The correct answer is (1/2)(1/7) + (1/2)(1/7) - (1/2)(1/7)(1/2)(1/7).
The rule you’re not following is:
P(A or B) = P(A) + P(B) - P(A and B)
When these mistakes are corrected, the correct answer comes out:
This is correct, if you are excluding the case where both are boys both born on Tuesday. Otherwise you would not subtract p(A and B). But, you did not say only one, you said _at least_ one.
I don’t think you have the dependencies quite right, because you can actually use more of the information than you do above to restrict the population from which you draw.
The real underlying population you should draw on seems to to be the population of fathers with exactly two children, of which one might be a boy born on Tuesday.
p(a two boy family given one brother was born on Tuesday) = (p(one brother born on Tuesday in a in two-boy family)) (p(two boys in 2 person families)) / p(out of all two person families, having one be a boy born on Tuesday)
which is if we say Tuesday birth is 1⁄7 and boy is 1⁄2,
(2/7) (1/4) / (2/14) = 1⁄2 so the Tuesday datum drops out.
You’re not stating what probability rules (theorems/axioms) you are using (you’re probably going by intuition), and you have made mistakes.
p(one brother born on Tuesday in a in two-boy family)is not 2⁄7; It’s1/7 + 1/7 - (1/7)(1/7)because you’re counting the two children both being born on Tuesday twice. The same mistake has been made in calculatingp(out of all two person families, having one be a boy born on Tuesday); The correct answer is(1/2)(1/7) + (1/2)(1/7) - (1/2)(1/7)(1/2)(1/7).The rule you’re not following is:
P(A or B) = P(A) + P(B) - P(A and B)When these mistakes are corrected, the correct answer comes out:
((1/7 + 1/7 - (1/7)*(1/7))*1/4)/((1/2)*(1/7) + (1/2)*(1/7) - (1/2)*(1/7)*(1/2)*(1/7)) = 13/27 =~ 0.4814This is correct, if you are excluding the case where both are boys both born on Tuesday. Otherwise you would not subtract p(A and B). But, you did not say only one, you said _at least_ one.
It’s not about excluding that case. It’s about not counting it twice. Search for the inclusion-exclusion principle to see the reasoning behind it.