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.
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.