Crates of eggs are inspected for blood clots by randomly sel

Crates of eggs are inspected for blood clots by randomly selecting 5 eggs from the crate without replacement and examining their contents. Let Y be the number of eggs containing blood clots out of the 5 inspected

If the crate contains 60 eggs, of which 10 have blood clots, what is the probability three or more of the 5 eggs inspected will contain blood clots?

Solution

Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial
crate contains 60 eggs, of which 10 have blood clots
p = 10/60 = 0.1667 ; q = 1 - 0.1667 = 0.8333
P( X > = 3 ) = 1 - P( X < 3) = 0.036
P( X < 3) = P(X=2) + P(X=1) + P(X=0)   
= ( 5 2 ) * 0.1667^2 * ( 1- 0.1667 ) ^3 + ( 5 1 ) * 0.1667^1 * ( 1- 0.1667 ) ^4 + ( 5 0 ) * 0.1667^0 * ( 1- 0.1667 ) ^5
= 0.964