M838TE wrote:
IanStewartcould you or someone help explain your approach on 1- P(not harry).
Oh, that certainly wouldn't be "my approach" to this problem, as I hope I made clear. I said I "wouldn't recommend" this kind of approach to this problem, but perhaps I should have expressed that differently. Using combinatorics formulas here turns a simple problem into a needlessly complicated one.
I was only discussing that approach because kornn directed a question to me about why the method they were using didn't seem to work, so it was their approach that I was talking about, and I would never consider using those combinatorics formulas to answer a question like this. The answer to this question is instantly 2/10 if you look at it in the right way. If you think about this question:
• 10 people, including Harry, line up in a random order. What is the probability Harry is 3rd in line?
then the answer is 1/10, because Harry is equally likely to be in any position. If you then think of this question:
• 10 people, including Harry, line up in a random order. What is the probability Harry is 2nd or 3rd in line?
then the answer is 2/10, because in 2 of the 10 spots we can put Harry, he's 2nd or 3rd in line. If you then think about this question:
• 10 people, including Harry, line up in a random order, and the person 2nd in line will be called the "Treasurer" and the person third in line will be called the "Secretary". What is the probability Harry will be Treasurer or Secretary?
then I haven't changed the problem at all, so the answer is still 2/10.
All of that said, if you did want to understand why the answer to this question is also equal to 1 - (9C2 / 10C2) (and to reiterate, I'd never consider solving the problem this way), then it's important to understand what "9C2" and "10C2" mean. 10C2 means "the number of groups of two you can pick from a group of 10", where we don't care about order. So from the 10 people, there are 10C2 pairs of people we could choose to fill the secretary and treasurer positions, if we don't care which person is in which role. If we know we do not want Harry to occupy either of those roles, then we have fewer ways to fill them: we then need to fill the 2 roles from only 9 people. So we then have 9C2 pairs of people we could choose to fill the secretary and treasurer roles (again if order does not matter). So when we pick just two people to fill those two roles, the probability is 9C2/10C2 that Harry is
not selected, and therefore 1 - 9C2/10C2 is the probability Harry
is selected. But that's a very complicated way to solve the problem.
thank you so much for the detail explanation. this (order/non-order or repeatability) is what I mean by I learned something from this problem. Just wanted to confirm my understanding. thanks for the detail explanation.