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Problem description:

A group of 5 boys and 10 girls is lined up in random order -- that is, each of the 15! permutations is assumed to be equally likely.

  1. What is the probability that the person in the 4th position is a boy?

  2. What is the probability that a particular boy is in the 3rd position?

I don't really know how to reason about problems like this. I feel like the answer to part 1 is 14!*5. I can select one of 5 boys in the fourth position and then I have 14 people left to arrange in the remaining spots. My best guess part 2 is just 14! where one position is occupied by a particular boy and the rest of the 14 people are then arranged in the spots.

I don't understand how to think about these types of problems and I have this constant, nagging feeling of something important being left out from my reasoning.

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HINT: For the first question, simply observe that each of the $15$ people is equally likely to be in the fourth position. For the second, observe that the boy is equally likely to occupy any of the $15$ positions.

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  • $\begingroup$ ok so it's 1/15 for a random person to be in the fourth position and there are 5 boys so 5/15? And just 1/15 for the second? $\endgroup$ – Andy Apr 18 '15 at 18:39
  • $\begingroup$ @Andy: That’s right. (This is a problem in which it’s easy to lose oneself in extraneous detail.) $\endgroup$ – Brian M. Scott Apr 18 '15 at 18:42
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Your reasoning is right. The 4th position can be occupied by any 5 boys and rest 14 positions by remaining people in 14! ways. Hence 5*14!

Since we have to find probability you will do; P(Event)/P(sample space).

Hence: 5*14!/15! which is 5/15.

Same for the 2nd part too.

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You're correct that there are $15!$ different permutations overall, but a lot of them are equivalent for your problems. You can count all the possible solutions (including all permutations) and divide that by $15!$, but there's an easier way to think about it.

  1. What is the probability that a randomly picked child is a boy.
  2. Given a particular boy and randomly choosing a position to assign him to, what is the probability that it's the 3rd position.
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