A bag has 1000 balls, some colored red, and the rest green. The fraction of red-colored balls in the bag is 0.2. A blindfolded observer picks out 5 balls from this bag. What is the probability that 3 reds are picked?
Method 1:
The probability p of picking a red ball is 0.2. So, this can be treated as a binomial problem:
$ Pr$(3 red balls picked) $= {5 \choose 3} (0.2)^3 (0.8)^2 = 0.0512$
Method 2:
We choose 3 red balls from a group of 200 and 2 green ones from the group of 800. So, the probability becomes:
$ \frac{{200 \choose 3} {800 \choose 2}} {{1000 \choose 5}} \approx 0.0508785...$
Why is there a slight discrepancy in the probabilities? Is it an artifact of my calculator's limitations or am I missing something deeper?