I created six hundred coins. I tell you that each is red on one side, but may be red or blue on the other side. I flip each coin, and show you the resulting colors. You count 400 red and 200 blue. What is your best estimate of the number of coins that are red on both sides?
I flipped all the same coins again, and you count 350 red and 250 blue. How should you modify your estimate?
(In reply to I might think you cheated.
"The probability of 200 or fewer blue occurring when these are all
flipped is (using the notation of a TI-83 calculator) binomcdf(n,.5,200)"
But you don't want this -- you want the probability of getting EXACTLY 200 blues.