Quantitative Methods for Business Cha 2—Introduction to Probability PROBLEM
Chapter 2—Introduction to Probability
PROBLEM
1. A market study taken at a local sporting goods store showed that of 20 people questioned, 6 owned tents, 10 owned sleeping bags, 8 owned camping stoves, 4 owned both tents and camping stoves, and 4 owned both sleeping bags and camping stoves.
Let:
Event A = owns a tent
Event B = owns a sleeping bag
Event C = owns a camping stove
and let the sample space be the 20 people questioned.
a.
Find P(A), P(B), P(C), P(A C), P(B C).
b.
Are the events A and C mutually exclusive? Explain briefly.
c.
Are the events B and C independent events? Explain briefly.
d.
If a person questioned owns a tent, what is the probability he also owns a camping stove?
e.
If two people questioned own a tent, a sleeping bag, and a camping stove, how many own only a camping stove? In this case is it possible for 3 people to own both a tent and a sleeping bag, but not a camping stove?
ANSWER:
a.
P(A) = .3; P(B) = .5; P(C) = .4; P(A B) = .2; P(B C) = .2
b.
Events B and C are not mutually exclusive because there are people (4 people) who both own a tent and a camping stove.
c.
Since P(B C) = .2 and P(B)P(C) = (.5)(.4) = .2, then these events are independent.
d.
.667
e.
Two people own only a camping stove; no, it is not possible
2. An accounting firm has noticed that of the companies it audits, 85% show no inventory shortages, 10% show small inventory shortages and 5% show large inventory shortages. The firm has devised a new accounting test for which it believes the following probabilities hold:
P(company will pass test | no shortage)
= .90
P(company will pass test | small shortage)
= .50
P(company will pass test | large shortage)
= .20
a.
If a company being audited fails this test, what is the probability of a large or small inventory shortage?
b.
If a company being audited passes this test, what is the probability of no inventory shortage?
ANSWER:
a.
.515
b.
.927
PROBLEM
1. A market study taken at a local sporting goods store showed that of 20 people questioned, 6 owned tents, 10 owned sleeping bags, 8 owned camping stoves, 4 owned both tents and camping stoves, and 4 owned both sleeping bags and camping stoves.
Let:
Event A = owns a tent
Event B = owns a sleeping bag
Event C = owns a camping stove
and let the sample space be the 20 people questioned.
a.
Find P(A), P(B), P(C), P(A C), P(B C).
b.
Are the events A and C mutually exclusive? Explain briefly.
c.
Are the events B and C independent events? Explain briefly.
d.
If a person questioned owns a tent, what is the probability he also owns a camping stove?
e.
If two people questioned own a tent, a sleeping bag, and a camping stove, how many own only a camping stove? In this case is it possible for 3 people to own both a tent and a sleeping bag, but not a camping stove?
ANSWER:
a.
P(A) = .3; P(B) = .5; P(C) = .4; P(A B) = .2; P(B C) = .2
b.
Events B and C are not mutually exclusive because there are people (4 people) who both own a tent and a camping stove.
c.
Since P(B C) = .2 and P(B)P(C) = (.5)(.4) = .2, then these events are independent.
d.
.667
e.
Two people own only a camping stove; no, it is not possible
2. An accounting firm has noticed that of the companies it audits, 85% show no inventory shortages, 10% show small inventory shortages and 5% show large inventory shortages. The firm has devised a new accounting test for which it believes the following probabilities hold:
P(company will pass test | no shortage)
= .90
P(company will pass test | small shortage)
= .50
P(company will pass test | large shortage)
= .20
a.
If a company being audited fails this test, what is the probability of a large or small inventory shortage?
b.
If a company being audited passes this test, what is the probability of no inventory shortage?
ANSWER:
a.
.515
b.
.927