4.14: Solutions
- Page ID
- 5559
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| \(x\) | \(P(x)\) |
|---|---|
| 0 | 0.12 |
| 1 | 0.18 |
| 2 | 0.30 |
| 3 | 0.15 |
| 4 | 0.10 |
| 5 | 0.10 |
| 6 | 0.05 |
3. 0.10 + 0.05 = 0.15
5. 1
7. 0.35 + 0.40 + 0.10 = 0.85
9. 1(0.15) + 2(0.35) + 3(0.40) + 4(0.10) = 0.15 + 0.70 + 1.20 + 0.40 = 2.45
11.
| \(x\) | \(P(x)\) |
|---|---|
| 0 | 0.03 |
| 1 | 0.04 |
| 2 | 0.08 |
| 3 | 0.85 |
13. Let \(X =\) the number of events Javier volunteers for each month.
15.
| \(x\) | \(P(x)\) |
|---|---|
| 0 | 0.05 |
| 1 | 0.05 |
| 2 | 0.10 |
| 3 | 0.20 |
| 4 | 0.25 |
| 5 | 0.35 |
17. 1 – 0.05 = 0.95
18. \(X =\) the number of business majors in the sample.
19. 2, 3, 4, 5, 6, 7, 8, 9
20. \(X =\) the number that reply “yes”
22. 0, 1, 2, 3, 4, 5, 6, 7, 8
24. 5.7
26. 0.4151
28. \(X =\) the number of freshmen selected from the study until one replied "yes" that same-sex couples should have the right to legal marital status.
30. 1,2,…
32. 1.4
35. 0, 1, 2, 3, 4, …
37. 0.0485
39. 0.0214
41. \(X =\) the number of U.S. teens who die from motor vehicle injuries per day.
43. 0, 1, 2, 3, 4, ...
45. No
48.
- X = the number of pages that advertise footwear
- 0, 1, 2, 3, ..., 20
- 3.03
- 1.5197
50.
- X = the number of Patriots picked
- 0, 1, 2, 3, 4
- Without replacement
53. \(X =\) the number of patients calling in claiming to have the flu, who actually have the flu. X = 0, 1, 2, ...25
55. 0.0165
57.
- X = the number of DVDs a Video to Go customer rents
- 0.12
- 0.11
- 0.77
59. 4.43
61. c
63.
- X = number of questions answered correctly
- X ~ B(32, 13)(32, 13)
- We are interested in MORE THAN 75% of 32 questions correct. 75% of 32 is 24. We want to find P(x > 24). The event "more than 24" is the complement of "less than or equal to 24."
- P(x > 24) = 0
- The probability of getting more than 75% of the 32 questions correct when randomly guessing is very small and practically zero.
65.
- X = the number of college and universities that offer online offerings.
- 0, 1, 2, …, 13
- X ~ B(13, 0.96)
- 12.48
- 0.0135
- P(x = 12) = 0.3186 P(x = 13) = 0.5882 More likely to get 13.
67.
- X = the number of fencers who do not use the foil as their main weapon
- 0, 1, 2, 3,... 25
- X ~ B(25,0.40)
- 10
- 0.0442
- The probability that all 25 not use the foil is almost zero. Therefore, it would be very surprising.
69.
- X = the number of audits in a 20-year period
- 0, 1, 2, …, 20
- X ~ B(20, 0.02)
- 0.4
- 0.6676
- 0.0071
71.
- X = the number of matches
- 0, 1, 2, 3
- In dollars: −1, 1, 2, 3
- 1212
- The answer is −0.0787. You lose about eight cents, on average, per game.
- The house has the advantage.
73.
- X ~ B(15, 0.281)
-
- Mean = μ = np = 15(0.281) = 4.215
- Standard Deviation = σ = npq−−−√𝑛𝑝𝑞 = 15(0.281)(0.719)−−−−−−−−−−−−−√15(0.281)(0.719) = 1.7409
- P(x > 5)=1 – 0.7754 = 0.2246
P(x = 3) = 0.1927
P(x = 4) = 0.2259
It is more likely that four people are literate that three people are.
75.
- X = the number of adults in America who are surveyed until one says he or she will watch the Super Bowl.
- X ~ G(0.40)
- 2.5
- 0.0187
- 0.2304
77.
- X = the number of pages that advertise footwear
- X takes on the values 0, 1, 2, ..., 20
- X ~ B(20, 2919229192)
- 3.02
- No
- 0.9997
- X = the number of pages we must survey until we find one that advertises footwear. X ~ G(2919229192)
- 0.3881
- 6.6207 pages
79. 0, 1, 2, and 3
81.
- X ~ G(0.25)
-
- Mean = μ = 1p1𝑝 = 10.2510.25 = 4
- Standard Deviation = σ = 1−pp2−−−√1−𝑝𝑝2 = 1−0.250.252−−−−−√1−0.250.252 ≈ 3.4641
- P(x = 10) = 0.0188
- P(x = 20) = 0.0011
- P(x ≤ 5) = 0.7627
82.
- X ~ P(5.5); μ = 5.5; σ = 5.5−−−√𝜎 = 5.5 ≈ 2.3452
- P(x ≤ 6) ≈ 0.6860
- There is a 15.7% probability that the law staff will receive more calls than they can handle.
- P(x > 8) = 1 – P(x ≤ 8) ≈ 1 – 0.8944 = 0.1056
84.
Let X = the number of defective bulbs in a string.
Using the Poisson distribution:
- μ = np = 100(0.03) = 3
- X ~ P(3)
- P(x ≤ 4) ≈ 0.8153
Using the binomial distribution:
- X ~ B(100, 0.03)
- P(x ≤ 4) = 0.8179
The Poisson approximation is very good—the difference between the probabilities is only 0.0026.
86.
- X = the number of children for a Spanish woman
- 0, 1, 2, 3,...
- 0.2299
- 0.5679
- 0.4321
88.
- X = the number of fortune cookies that have an extra fortune
- 0, 1, 2, 3,... 144
- 4.32
- 0.0124 or 0.0133
- 0.6300 or 0.6264
- As n gets larger, the probabilities get closer together.
90.
- X = the number of people audited in one year
- 0, 1, 2, ..., 100
- 2
- 0.1353
- 0.3233
92.
- X = the number of shell pieces in one cake
- 0, 1, 2, 3,...
- 1.5
- 0.2231
- 0.0001
- Yes
94. d