Question

An urn contains 1000 balls, 825 of which are blue and the rest are red. I sample 100 balls. What is the probability that I pick exactly 80 blue balls in my sample, if:

(a) I sample without replacement? (For you to check your work, the anwer is approximately 0.0834325).

(b) I sample with replacement? (This time, the answer is approximately 0.0805802).

(c) Can you explain why your number in part a is bigger than your number in part b?

Answer 1

a) required probability = P(X=80)= C(825,80)*C (175,20)/C(1000,100)           =   0.083433

b)

n=100

p=825/1000=0.825

X~Bin(100,0.825)

P ( X =    80   ) = C(100,80) * 0.825^80 * (1-0.825)^20 =        0.0805802   (answer)

c)

because in part a) without replacement, probability of choosing blue balls increases after each ball is drawn

but in part b) probability of choosing blue balls remains same

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