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