Bag full of money

[mindless]

Someone gives you a bag full of 100 checks. The checks have amounts equal to $1, $2...all the way up to $100. You are allowed to pull out one check at a time. If you are happy with that amount, you may keep it and stop playing. If not, you may put the check back in the bag and pick another at random. If you put the 9th check bag in the bag, you *must* keep the 10th check that you take.

A) What strategy should you use to maximize your expectation?
B) What is your expectation?

[hmkpoker]

Not that I don't like the OP, but this belongs in the Puzzles and Other Games forum.

[hmkpoker]

If the selected check number is greater than (100 - (100/(11-n))) where n = the number of the trial, you keep the check, else you dump it.

Expectation starts at $90 on the first trial, and declines with every trial thereafter. In the event that a selected check is less than expectation but greater than expectation on the next trial, it's probably necessary to still keep it anyway since it's above the expectation of what you'll have if you dump it.

I'm sure there's some minor adjustment that can be made to compensate for the displaced checks, but I don't really feel like figuring that out.