Choosing K balls out of N balls in 2 boxes. Likelihood that at least M are from box 2..

Question

Here is the problem I am facing.

I have 1,000 balls, of which 970 are red and 30 are blue.

I randomly choose 950 balls, without replacement.

What are the odds that at least 930 are red?

I know there are (1,000 choose 950) ways choose the balls. But I am not sure how to get how many of those choices have at least 930 red balls.

Thank you very much in advance.

Answer 1

HINT You can't have more than $20$ blue, so find the probability to have $0$ up to $20$ blue.

The probability to have exactly $k$ blue is $\binom{970}{950-k}\binom{30}{k}$ and these events are independent, so you can sum the probabilities.