Thread: Conditinal Probability with numbers selected at random

Suppose that three numbers are selected one by one, at random, and without replacement from the set of numbers {1, 2, 3, ... n}. What is the probability that the third number falls between the first two if the first is smaller than the second?

The book tells me that the answer is 1/3, but that isn't much help towards how to find the answer.

Hello, Zennie!

Let the three numbers be: .

There are permutations of the three numbers:
. . . (in increasing order)


"The first is less than the second": .
. . There are three cases: .

In only one case, is between and : .

Therefore, the probability is: .

Wow, thanks a lot. That's so much more simple than I was expecting. I probably wouldn't have thought about it that way.

It is not such a tough question. Try using this hint:
Pick three number (in order) - x1,x2,x3 from {1,2,3,.....,n}
What is the Prob(x1<x2<x3)? [You don't need any calculation - think symmetry !!]