Probability of being highest OR second highest draw from a distribution

Question

In my situation, I have a distribution F(x) over some compact interval. Say I take $n$ iid draws from the distribution. I want to find the probability that one draw, $x_i$, is the highest of the $n$ draws OR is the second highest of the $n$ draws.

If it were just finding the probability of being the highest, it would just be $F(x_i)^{n-1}$, I believe.

And I believe that the probability of being exactly the second highest is ${n-1 \choose n-2} F(x_i)^{n-2} (1-F(x_i))$.

But when I'm trying to find the joint probability, I feel like I might be missing some overlap if I just add those together.

Thanks!

EDIT: Maybe I phrased my question incorrectly. I'm looking at an auction situation, where every bidder gets an independent draw as his value in the auction. Each bidder makes a bid & then the highest bidder wins the item and pays his bid. So I'm trying to find any bidder's expected profit, which would be his value minus his bid, but only in the case that he's the highest bidder. So I thought it would be (v−b)F(b)n−1, where F(b)n−1 is the "probability of being the highest," as I phrased it. But I was trying to do the case where both highest & second highest win

Answer 1

I will assume that the distribution is continuous, so that the probability of ties is $0$.

The probability that a specific draw, say the seventh, is the highest is $\frac{1}{n}$. This is because all draws are equally likely to be highest. The probability it is second highest is $\frac{1}{n}$. Add.

Answer 2

The reason that André Nicolas's solution is correct, and yours incorrect, is that you are confusing the ordinal ranking of the draws, which he addresses, and is what you asked for, with the (cunulative) distribution function of the value of the order statistics x1 and x2. He addresses the probability that x1 will be highest or 2nd highest. You are providing formulas for the distribution function (hence value) of the winner (1st place) or runner-up (2nd place).

It's like the difference between predicting which day in the year will get the most rain vs. what the greatest amount of rain to fall on any day will be.