A mathmatical question(mistake/paradox)about the probability.

Question

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.)

Answer 1

Ok lets say A happens if a person is under 45 and B happens if the person is older than 45 years.
For picking two persons we have four scenarios:
AA AB BA BB

BB can be excluded because we must have at least one person of type A.
BA can also be excluded in case the order in which the persons are picked does matter.

AA has 6 * 5 = 30 possible combinations.
AB has 6 * 2 = 12 possible combinations.
The amount of overall combinations is therefore 30 + 12 = 42.
P(AB) = 12 / 44 = 2/7

In case the order doesn't matter and simply at least one of the two picked persons has to be of type A, BA is also a valid combination.

BA has 6 * 2 = 12 possible combinations.
The total amount of combinations is therefore increased by 12 to 54.
P(AB) + P(BA) = 12 / 54 + 12 / 54 =
= 24 / 54 = 4/9

Answer 2

Well- it is a good solution, to directly talk about the numbers instead of the possibility.
Well I also need someone to defeat that second solution. Thanks!

Answer 3

This question is not programming related so it's most likely going to be marked for deletion.

That said, the problem is conditional probability doesn't care about order, so you're not only calculating P(scenario AB), but also P(scenario BA), where A is a person under 45 and B is a person over 45.

Both scenarios have the same probability, just divide by two get P(scenario AB).

Answer 4

Well, OK i didn't think about this..Thank you