Chapter 7 – Probability Distribution Answer Key

CK-12 Probability and Statistics Concepts 1

7.1 Understanding Discrete Random Variables

Answers

1. K is a random variable because it is the numerical result of a limited number of independent trials.

2. S is not a random variable because there is not a limited number of trials described in the

experiment. 3. H is a random variable. 4. T is a random variable. 5. B is a random variable. 6. S is not a random variable, the outcome of the experiment is not numerical. 7. N is not a random variable, it only has a single possible outcome, and it is not random. 8. A is a random variable. 9. H is not random, there is only one possible outcome and it is not determined by chance. 10. H is a random variable.

Chapter 7 – Probability Distribution Answer Key

CK-12 Probability and Statistics Concepts 2

7.2 Understanding Continuous Random Variables

Answers

1. Continuous 2. Discrete 3. Continuous 4. Discrete 5. Discrete 6. Neither (not a numeric value, so not a random variable) 7. Continuous 8. Discrete 9. Discrete 10. Discrete

Chapter 7 – Probability Distribution Answer Key

CK-12 Probability and Statistics Concepts 3

7.3 Probability Distribution

Answers

1. A probability distribution is a list of each value a random variable can attain, along with the probability of attaining each value.

2. A random variable assigns a unique numerical value to each output of a random process. 3. A discrete random variable has a countable number of possible values. A continuous variable

can take on any value in an interval. 4. 5 sides 5. .08 6. .2 7. .08 8. Appearance of answers may vary, but should contain the following values:

𝑆 2 3 4 5 6 7 8 9 10 11 12 13 14

𝑃(𝑆) .0204 .0408 .0612 .0816 .102 .1224 .1429 .1224 .102 .0816 .0612 .0408 .0204

9. Appearance of answers may vary, but should contain the following values:

𝐻 0 1 2 3

𝑃(𝐻) .125 .375 .375 .125

10. Appearance of answers may vary, but should contain:

𝑆 2 3 4 5 6 7 8 9 10 11 12

𝑃(𝑆) .0278 .0556 .0833 .1111 .1389 .1667 .1389 .1111 .0833 .0556 .0278

Chapter 7 – Probability Distribution Answer Key

CK-12 Probability and Statistics Concepts 4

7.4 Visualizing Probability Distribution

Answers

1. 1 roll of two 7-sided dice: 2 or 14 3 or 13 4 or 12 5 or 11 6 or 10 7 or 9 8 0.0204 0.0408 0.0612 0.0816 0.102 0.1224 0.1429

2.

3.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

2 3 4 5 6 7 8 9 10 11 12 13 14

Roll Outcome

Roll Outcome

Roll Outcome

2 3 4 5 6 7 8 9 10 11 12 13 14

Chapter 7 – Probability Distribution Answer Key

CK-12 Probability and Statistics Concepts 5

4. Probability distribution of 𝑅: 𝑅 0 1 2 3 4

𝑃(𝑅) 59% 9.5% 2% 0.3% .01% 5.

6.

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

0 1 2 3 4

Number of Red

Number of Red

Number of Red

0 1 2 3 4

Chapter 7 – Probability Distribution Answer Key

CK-12 Probability and Statistics Concepts 6

7. Probability distribution for 𝑆: 𝑆 0 1 2 3

𝑃(𝑆) (

21

25)

3= .593

=59.3%

(4

25) × (

21

25)

2×

3 =.3386 =33.86%

(4

25)

2×

(21

25) × 3

=.065 =6.5%

(4

25)

3=.004=.4%

8.

9.

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

0 1 2 3

Number of Sevens

Number of Sevens

Number of Sevens

0 1 2 3

Chapter 7 – Probability Distribution Answer Key

CK-12 Probability and Statistics Concepts 7

10. Probability distribution for 𝑇: 𝑇 0 1 2 3 4

𝑃(𝑇) .0625 .25 .375 .25 .0625 11. (may also be a histogram)

12.

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

35.00%

40.00%

0 1 2 3 4

Number of Tails

Number of Tails

Number of Tails

0 1 2 3 4

Chapter 7 – Probability Distribution Answer Key

CK-12 Probability and Statistics Concepts 8

7.5 Probability Density Functions

Answers

1. 55% 2. 100% 3. 45% 4. 0% 5. 27% 6. 0% 7. 73% 8. 100% 9. 68% 10. 81.5% 11. 97.5% 12. 97.5% 13. 2.35% 14. .15% 15. 0% (effectively)

Chapter 7 – Probability Distribution Answer Key

CK-12 Probability and Statistics Concepts 9

7.6 Binomial Experiments

Answers

1. Binomial 2. Not binomial, probability changes each trial (candy is eaten) 3. Not binomial, outcomes are not success/failure 4. Not binomial, number of trials is not fixed 5. Binomial 6. Not binomial, outcome is a number, not success/failure 7. Binomial 8. Not binomial, number of trials not fixed 9. Binomial 10. Binomial 11. Not binomial, number of trials not fixed 12. Binomial

Chapter 7 – Probability Distribution Answer Key

CK-12 Probability and Statistics Concepts 10

7.7 Expected Value

Answers

1. .3*4.1+.45*4.4+.1*4.7+.05*4.9+.1*5.1 = 4.435 2. 4*.5+8*.25+12*.15+16*.05+20*.05 = 7.6 3. 15*.2+30*.25+45*.15+60*.27+75*.13 = 43.2 4. 30*.18+60*.16+90*.24+120*.22+150*.20 = 93 5. 3*.07+11*.08+19*.65+27*.2 = 18.84 6. 13*.15+17*.17+21*.23+25*.30+29*.10+33*.03+37*.02 = 21.80 7. 26*.06+39*.14+52*.3+65*.28+78*.22 = 57.98 8. 22*.105+43*.225+64*.315+85*.228+106*.127 = 64.987 9. .65*.16+.84*.29+1.03*.14+1.22*.28+1.41*.13 = 1.0167

10. 3

8∗

2

5+

1

2∗

1

4+

5

8∗

1

5+

3

4∗

1

10+

7

8∗

1

20=

83

160= .51875

11. 20*.15+25*.35+30*.3+35*.15+40*.05 = 28 shines per day * $3.50 = $98 per day 12. 20*.15+25*.28+30*.32+35*.15+40*.1 = 28.85 avg hours * $8.50 = $245.23 per week *1/3 = He

can spend $81.75 per week 13. 28.85 avg hours * $10.50 per hour = $302.93 *1/3 = $101 per week

Chapter 7 – Probability Distribution Answer Key

CK-12 Probability and Statistics Concepts 11

7.8 Random Variable Variance

Answers

1. ó𝟐𝑿 =. 𝟎𝟗𝟔, ó𝑿 =. 𝟑𝟏

2. ó𝟐

𝑿 = 𝟐𝟎. 𝟔𝟒, ó𝑿 = 𝟒. 𝟓𝟒 3. ó𝟐

𝑿 = 𝟒𝟏𝟎. 𝟕𝟔, ó𝑿 = 𝟐𝟎. 𝟐𝟕 4. ó𝟐

𝑿 = 𝟏𝟕𝟎𝟏, ó𝑿 = 𝟒𝟏. 𝟐𝟒 5. ó𝟐

𝑿 = 𝟖. 𝟗𝟓, ó𝑿 = 𝟐. 𝟗𝟗 6. ó𝟐

𝑿 = 𝟑𝟐. 𝟑𝟐, ó𝑿 = 𝟓. 𝟔𝟗 7. ó𝟐

𝑿 = 𝟐𝟐𝟒. 𝟓, ó𝑿 = 𝟏𝟒. 𝟗𝟖 8. ó𝟐

𝑿 = 𝟔𝟎𝟖, ó𝑿 = 𝟐𝟒. 𝟔𝟓 9. ó𝟐

𝑿 = 𝟔. 𝟒𝟎, ó𝑿 = 𝟐. 𝟓𝟑 10. ì𝑿 = 𝟑𝟐. 𝟐𝟑 𝒉𝒐𝒖𝒓𝒔, ó𝟐

𝑿 = 𝟐𝟐. 𝟑𝟒, ó𝑿 = 𝟒. 𝟕𝟑 = $𝟓𝟒. 𝟒𝟎 He cannot afford the payment, 25% of his avg monthly is $370.65

Chapter 7 – Probability Distribution Answer Key

CK-12 Probability and Statistics Concepts 12

7.9 Transforming Random Variables I

Answers

1. ì𝑋+.02 = .09 and ó𝑋+.02 = .002(unchanged) 2. ì𝐶+13.6 = 157.6 and ó𝐶+13.6 = 17(unchanged) 3. ì𝑌−3.49 = 18.51 and ó𝑌−3.49 = 1.8(unchanged) 4. ì𝑍+4.25 = 72.58 and ó𝑍+4.25 = 4.87(unchanged) 5. ì𝑋−1.035 = 1.036 and ó𝑋−1.035 = .807(unchanged)

6. ì𝐴+

3

10

=9

10𝑜𝑟. 9 and ó

𝐴+3

10

=1

5𝑜𝑟. 2(unchanged)

7. ì𝐵−.035 = 16.996 and ó𝐵−.035 = 2.101(unchanged)

8. ì𝐴−

3

14

=1

14 and ó

𝐴−3

14

=1

7(unchanged)

9. ì𝑌+.27 = 13.37 and ó𝑌+.27 = 3.01(unchanged) 10. ì𝑋+2 = 6 and ó𝑋+2 = .25 11. ì𝑋−1.25 = 9.25 and ó𝑋−1.25 = .75 12. ì𝑋−4 = 11 and ó𝑋−4 = 5 13. ì𝑋−.75 = 11.25 and ó𝑋−.75 = 1.5

Chapter 7 – Probability Distribution Answer Key

CK-12 Probability and Statistics Concepts 13

7.10 Transforming Random Variables II

Answers

1. ì3.1𝑋 = .713 and ó3.1𝑋 = 0.093 2. ì4𝐶 = 496 and ó4𝐶 = 48 3. ì−1.9𝑌 = −36.48 and ó−1.9𝑌 = −4.37 4. ì2.3𝑍 = 111.274 and ó2.3𝑍 = 5.221 5. ì−1.98𝑋 = −15.52 and ó−1.98𝑋 = −1.43

6. ì 3

14𝐴

=15

98 and ó 3

14𝐴

=3

98

7. ì−0.35𝐵+2 = −2.94 and ó−0.35𝐵+2 = −0.4025

8. ì2𝐴−

3

14

=9

14 and ó

2𝐴−3

14

=2

7

9. ì−3.8𝑌+2.13 = 83.07 and ó−3.8𝑌+2.13 = −11.172 10. ì3𝑋+2.25 = 14.25 and ó3𝑋+2.25 = 2.25 11. ì2𝑋−2.25 = 18.75 and ó2𝑋−2.25 = 19.5 12. ì3𝑋−4 = 41and ó3𝑋−4 = 15 13. ì2𝑍−.75 = 23.25 and ó2𝑍−.75 = 2.5