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Unformatted text preview: s A and B break down but not machine C or D.
Solution: P (A ∩ B ∩ C c ∩ Dc ) = P (A)P (B )P (C c )P (Dc ) = (.002)(.003)(1 − .001)(1 − .002) = .000005982
(E) Machine A breaks down but not machines B or C or D.
Solution: P (A ∩ B c ∩ C c ∩ Dc ) = P (A)P (B c )P (C c )P (Dc ) = (.002)(1 − .003)(1 − .001)(1 − .002) = .001988 7 Problem 10
The probability distribution of the random variable X is shown in the accompanying table.
X=x
10
5
0
5
10
15
20
P (X = x) 0.18 0.08 0.07 0.10 0.21 0.11 0.25
(A) Find P (X = −10).
Solution: P (X = −10) = .18
(B) Find P (X ≥ 5).
Solution: P (X ≥ 5) = .10 + .21 + .11 + .25 = .56
(C) Find P (0 ≤ X ≤ 15).
Solution: P (0 ≤ X ≤ 15) = .07 + .10 + .21 + .11 = .49
(D) Find E (X ).
Solution: E (X ) = −10(.18) + (−5)(.08) + 0(.07) + 5(.10) + 10(.21) + 15(.11) + 20(.25) = 7.05
Problem 11
Two cards are drawn from a wellshuﬄed deck of 52 playing cards. Let X denote the number of aces drawn.
(A) What are all the possible values X can take?
Solution: X can take the values 0, 1, or 2.
(B) Give the probability distribution of X in the form of a table.
X=x
Solution: 0
1
2 P ( X = x)
C (4,0)C (48,2)
C (52,2)
C (4,1)C (48,1)
C (52,2)
C (4,2)C (48,0)
C (52,2) = .8507
= .1448
= .0045 (C) What is P (X = 1)?
Solution: P (X = 1) = .1448
(D) What is E (X )?
Solution: E (X ) = 0(.8507) + 1(.1448) + 2(.0045) = .1538 8 Problem 12
After the private screening of a new television pilot, audience members were asked to rate the new show on a scale of 1 to 10
(10 being the highest rating). From a group of 140 people, the following responses were obtained.
Rating
1234
5
6
7
8
9 10
Frequency 2 3 4 12 22 17 21 22 12 25
Let the random variable X denote the rating given to the show by a randomly chosen audience member.
(A) Find the probability distribution of the random variable X .
Solution: X=x
P ( X = x) 1
2/140 2
3/140 3
4/140 4
12/140 5
22/140 6
17/140 7
21/140 8
22/140 9
12/140 (B) What is P (X = 5)?
Solution: P (X = 5) = 22
140 (C) What is P (X ≥ 8)?
Solution: P (X ≥ 8) = 22
140 + 12
140 + 25
140 = 59
140 ≈ .4214 (D) What rating can a viewer be expected to give this show?
2
3
4
12
22
17
21
22
12
25
E (X ) = 1( 140 ) + 2( 140 ) + 3( 140 ) + 4( 140 ) + 5( 140 ) + 6( 140 ) + 7( 140 ) + 8( 140 ) + 9( 140 ) + 10( 140 ) ≈ 6.86 9 10
25/140...
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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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