Question

From a shipment of 65 transistors, 4 of which are defective, a sample of 6 transistors is selected at random.
(a) In how many different ways can the sample be selected?
ways

(b) How many samples contain exactly 3 defective transistors?
166320 samples

(c) How many samples do not contain any defective transistors?
samples

Answer 1

Hypergeometric distribution applies to a limited population from which a sample is drawn without replacement.

Here we assume all defectives are identical, so are undefectives.
D=4 (defective)
U=65-4=61 (undefective) =>
S=U+D=65=size of population=65
d=defectives selected
u=undefectives selected =>
s=u+d=size of sample=6
C(n,r)=n!/(r!(n-r)!)=number of possible combinations selecting r from n objects.

(a)
Number of ways
=C(S,s)
=C(65,6)
=82598880

(b)
u=3, d=3
Number of samples with exactly 3 defectives
=C(D,d)*C(U,u)
=C(4,3)*C(61,3)
=143960

(c)
u=6
d=0
Number of samples without defectives
=C(D,d)*C(U,u)
=C(4,0)*C(61,6)
=55525372

Answer 2

From a shipment of 70 transistors, 4 of which are defective, a sample of 5 transistors is selected at random.