1
$\begingroup$

I'm looking for confirmation that what I for did this question is correct:

"If two people are randomly chosen from a group of eight women and six men, what is the probability that (a) both are women (b) one is a man and the other is a woman?"

a) The probability that both are women: $\frac{8}{14}\times\frac{7}{13}\approx.31$

b) The probability that one is a man and the other is a woman: $\frac{6}{14}\times\frac{8}{13}\approx.26$

$\endgroup$
  • $\begingroup$ Why did you divide by $16$ in $b$ ? $\endgroup$ – Belgi May 17 '14 at 17:24
  • $\begingroup$ oops, that was a typo. $\endgroup$ – Oscar Flores May 17 '14 at 17:25
1
$\begingroup$

Unfortunately, no. Your method works in part (a), but fails in part (b).

For a more generally workable method, we could proceed as follows. There are $\binom{14}2=91$ ways to pick $2$ people from a group of $14$. For part (a), there are $\binom82=28$ ways to pick two people from a group of $8$, so the answer is $\frac{28}{91}=\frac4{13},$ as you calculated.

For part (b), though, there are $8$ ways to choose one woman out of $8$ and $6$ ways to choose one man out of $6$, so there are $48$ ways to choose one man and one woman. Thus, the probability is $\frac{48}{91},$ instead.

Basically, what you found is the probability that a man was picked first and then a woman. However, for our purposes, we could have picked a woman first, instead, so we need to multiply this probability by $2$ to get the correct answer.

$\endgroup$
  • 2
    $\begingroup$ Why do you call (a) coincidence? His method is what I'd have used, and it's correct. $\endgroup$ – JoeTaxpayer May 17 '14 at 19:42
  • $\begingroup$ True. It's perfectly fine. $\endgroup$ – Cameron Buie May 17 '14 at 22:09
4
$\begingroup$

The first one is correct.

However, to see the problem with part $b$ try to answer this question: What is the probability that the first person is men and the second one is a woman ?

$\endgroup$
1
$\begingroup$

There are two possibilities of getting one of each gender:

First "man" then "woman" or first "woman" then "man"

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.