MEM 361 Engineering Reliability
Problem Set 2
1) Suppose that P{X}=0.32, P{Y}=0.44 and P{X∪ Y}=0.58. Answer the following questions
with proof.
(a) Are X and Y mutually exclusive?
(b) Are X and Y independent events?
(c) What is the value of P{X|Y}?
(d) What is the value of P{Y|X}?
Solution: a) ,
X
Y are mutually exclusive only if
(
)
0PX Y
∩
=. Now
()
(
)
(
)
(
)
()()()()
0.32 0.44 0.58 0.18
PX Y PX PY PX Y
PX Y PX PY PX Y
∪= + − ∩⇒
∩= + − ∪= + − =
So NOT mutually exclusive
b) For independent events
(
)
(
)
(
)
PX Y PXPY∩= , but this is not true, so NOT
independent
c)
()
()
(
)
0.18 / 0.44 0 .409PXY PX Y PY=∩ = =
d)
()
()
(
)
0.18 / 0.32 0.5625PYX P X Y PX=∩ = =
2) In a quality test of 126 computer chips from two different suppliers, the following are the test
results:
pass test fail test
Supplier -1 80 4
Supplier -2 40 2
Let A denote the event that a chip is from supplier-1 and B the event that a chip passes the quality
test. Answer the following questions.
(a) Are A and B independent events?
(b) Are Ac and B independent events?
(c) What is the meaning of P{A∪ B}?
(d) What is the value of P{A∪ B}?
Solution:
a)
() ()
(
)
()
()() ()
()
()() ()
84 / 126; 120 /126; 80 / 126
80 /120 2 / 3
80 / 84 20 / 21
PA PB PA B
PAB PA B PB PA
PBA PA B PA PB
==∩=
=∩ = ==
=∩ = = =
So ,
A
B ARE independent
b) ,
c
A
B ARE independent
c) The probability that A, B or both occur
d)
()()()()
84 120 80 124
126 126 126 126
PA B PA PB PA B∪= + − ∩= + − =
3) Use the Venn diagram and show the following equalities:
