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Unformatted text preview: heavy coﬀee drinker” and C be the event “having cancer of the pancreas.” Since we
are testing for independence, we must compare P (H ∩ C ) with P (H )P (C ). If they are equal then the events are independent.
If they are not equal then the events are not independent.
121
P (H ∩ C ) = 10,000 = .0121
P (H ) = 4,000
10,000 and P (C ) = 151
10,000 , so P (H )P (C ) = .00604. This means that P(H ∩ C) = P(H)P(C) and so H and C are NOT INDEPENDENT.
Problem 9
Copykwik has four photocopy machines A, B, C, and D. The probability that a given machine will break down on a particular
day is given below. Assuming independence, what is the probability on a particular day that the following will occur?
P (A) = .002 P (B ) = .003 P (C ) = .001 P (D) = .002 (A) None of the machines will break down.
Solution: This means A does not break down AND B does not break down AND C does not break down AND D does
not break down. So we want P (Ac ∩ B c ∩ C c ∩ Dc ). Since the events A, B , C , and D are independent we have
P ( Ac ∩ B c ∩ C c ∩ D c ) = P ( Ac ) P ( B c ) P ( C c ) P ( D c ) = (1 − P (A))(1 − P (B ))(1 − P (C ))(1 − P (D)) =
≈ (1 − .002)(1 − .003)(1 − .001)(1 − .002) .9920 (B) All of the machines break down.
Solution: P (A ∩ B ∩ C ∩ D) = P (A)P (B )P (C )P (D) = (.002)(.003)(.001)(.002) = 1.2 × 10−11
(C) Exactly one machine breaks down.
Solution: We want to ﬁnd
P ( A ∩ B c ∩ C c ∩ D c ) + P ( Ac ∩ B ∩ C c ∩ D c ) + P ( Ac ∩ B c ∩ C ∩ D c ) + P ( Ac ∩ B c ∩ C c ∩ D ) . P (A ∩ B c ∩ C c ∩ Dc ) = (.002)(1 − .003)(1 − .001)(1 − .002) = .001988021988
P (Ac ∩ B ∩ C c ∩ Dc ) = (1 − .002)(003)(1 − .001)(1 − .002) = .002985023988 P (Ac ∩ B c ∩ C ∩ Dc ) = (1 − .002)(1 − .003)(.001)(1 − .002) = .000993015988
P (Ac ∩ B c ∩ C c ∩ D) = (1 − .002)(1 − .003)(1 − .001)(.002) = .001988021988
So P ( A ∩ B c ∩ C c ∩ D c ) + P ( Ac ∩ B ∩ C c ∩ D c ) + P ( Ac ∩ B c ∩ C ∩ D c ) + P ( Ac ∩ B c ∩ C c ∩ D )
= .001988021988 + .002985023988 + .000993015988 + .001988021988
= .007954083952
≈ .007954
(D) Machine...
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This document was uploaded on 03/17/2014 for the course MATH 1331 at Texas Tech.
 Summer '06
 imnotsure
 Calculus, Probability

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