A mathmatical question(mistake/paradox)about the probability

Question:There are 8 people. Two of them are above 45 years old the others are under 45. Now we randomly choose two of them. What is the probability of choosing a person above 45 when we already know that another person is under 45?

Solution 1: We know that one of them is under 45so we take one from those six people. Now we have 2 above 45 and 5 under. So the probability is 2/7.

But someone write a solution with conditional probability: P(B|A)=P(AB)/P(A), to this situation,

P(Choose one under 45)=3/4

P(Choose one under 45 and one above 45)=(C(6,1)*C(2,1))/C(8,2)=3/7 So the solution is (3/7)/(3/4)=4/7

I can feel this solution have a mistake at the number 3/7 because it does't have any restriction on the sequence of choosing the people, but actually it has. The problem is, its mathematical expression seems flawless, I cannot find the mistake or write another correct one.

I would appreciate it if someone can find its problem in the math theory or write a correct one(Better with math expression.)

Note by Thomas Yao
4 weeks ago

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It's right, it's just Hypergeometric Distribution.

Pi Han Goh - 1 week, 3 days ago

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