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.)

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Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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

Pi Han Goh - 2 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$