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So everyone, I have a simple probability question that I cannot wrap my head around. It go like this:

There are $6$ men $9$ women on committee. $2$ people are chosen for treasurer and secretary. What is the probability one man is chosen for treasurer and one woman for secretary?

My solution goes like this:

$6$ men/$15$ people $\cdot$ $9$ women/$14$ people = $9/35$

but I was thinking, if we think that the denominator will be $15$ choose $2 = 105$, then we get different answer of $6 \cdot 9/105 = 18/35$???

Now I'm confused on which one is the correct answer. Someone help clear the confusion for me. Thanks.

N. F. Taussig
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Aiden Chow
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1 Answers1

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Let's call the "first" position treasurer and "second" position secretary. Then our order of choice matters in this setup.

Your initial computation (9/35) is correct since it computes the probability of putting a man in the first position and a woman in the second.

Your next computation (18/35) computes the ratio of the number of ways to choose a man and woman to the total number of ways to choose two people. But the numerator must be halved to account for the fact that only half of those ways place the man in the first position and woman in the second. This results in the correct answer.

Golden_Ratio
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  • thank for the answer, but the two position are chose at the same time, will that affect the answer? – Aiden Chow Jan 25 '22 at 09:41
  • Timing doesn't enter into the math. What matters is whether the choice is with or without replacement. In this case, it is without replacement, i.e. once you choose the first position, you have one less person in the pool to choose from. That's why your denominator is 14=15-1 in the second term of your initial computation. – Golden_Ratio Jan 25 '22 at 09:45
  • so the answer is same if for example secretary is chosen first? – Aiden Chow Jan 25 '22 at 09:48
  • if so, im little confused about that, it seem like the order should matter (my teacher always say order not matter and order matter are two different thing), but if you chose at same time, order not matter? explain that please, thank – Aiden Chow Jan 25 '22 at 09:50
  • The question doesn't make any mention of timing. By "order matters" here, I just meant the two positions are distinguishable (since choosing a man for "first" position is distinct from choosing a man for "second" one). If the positions are indistinguishable, 18/35 would be the correct answer. – Golden_Ratio Jan 25 '22 at 10:03
  • if i understanding, you saying that you have to divide 2 from 18/35 because man cant be chose for secretary? – Aiden Chow Jan 25 '22 at 10:13
  • @derppq exactly. – Golden_Ratio Jan 25 '22 at 10:15
  • thank, i understanding it now! – Aiden Chow Jan 25 '22 at 10:20
  • @derppq ok great! – Golden_Ratio Jan 25 '22 at 10:25