Summaries

13th September 2010

Session 1.7

 

Marginal, Joint and Conditional Probabilities

 

Suppose that we have fair dice: d4 with face values {1,2,3,4}, d6 with face values {1,2,3,4,5,6} and d8 with face values {1,2,3,4,5,6,7,8}.  Our experiment consists of first randomly selecting one of the dice and then tossing that die and noting the face value.

 

The first stage probabilities:

 

Pr{Select d4} = 1/3( = P4)

Pr{Select d6} = 1/3( = P6)

Pr{Select d8} = 1/3( = P8)

 

The conditional probabilities:

 

Pr{1 shows | d4 selected} = 1/4

Pr{2 shows | d4 selected} = 1/4

Pr{3 shows | d4 selected} = 1/4

Pr{4 shows | d4 selected} = 1/4

 

Pr{1 shows | d6 selected} = 1/6

Pr{2 shows | d6 selected} = 1/6

Pr{3 shows | d6 selected} = 1/6

Pr{4 shows | d6 selected} = 1/6

Pr{5 shows | d6 selected} = 1/6

Pr{6 shows | d6 selected} = 1/6

 

Pr{1 shows | d8 selected} = 1/8

Pr{2 shows | d8 selected} = 1/8

Pr{3 shows | d8 selected} = 1/8

Pr{4 shows | d8 selected} = 1/8

Pr{5 shows | d8 selected} = 1/8

Pr{6 shows | d8 selected} = 1/8

Pr{7 shows | d8 selected} = 1/8

Pr{8 shows | d8 selected} = 1/8

 

The joint probabilities

 

Pr{1 shows} = Pr{1 shows | d4 selected}*Pr{d4 selected} + Pr{1 shows | d6 selected}*Pr{d6 selected} + Pr{1 shows | d8 selected}*Pr{d8 selected} = (1/3)*(1/4) + (1/3)*(1/6) + (1/3)*(1/8) = (1/3)*(13/24) = 13/72 ≈ 0.1806

 

Pr{2 shows} = Pr{2 shows | d4 selected}*Pr{d4 selected} + Pr{2 shows | d6 selected}*Pr{d6 selected} + Pr{2 shows | d8 selected}*Pr{d8 selected} = (1/3)*(1/4) + (1/3)*(1/6) + (1/3)*(1/8) = (1/3)*(13/24) = 13/72 ≈ 0.1806

 

Pr{3 shows} = Pr{3 shows | d4 selected}*Pr{d4 selected} + Pr{3 shows | d6 selected}*Pr{d6 selected} + Pr{3 shows | d8 selected}*Pr{d8 selected} = (1/3)*(1/4) + (1/3)*(1/6) + (1/3)*(1/8) = (1/3)*(13/24) = 13/72 ≈ 0.1806

 

Pr{4 shows} = Pr{4 shows | d4 selected}*Pr{d4 selected} + Pr{4 shows | d6 selected}*Pr{d6 selected} + Pr{4 shows | d8 selected}*Pr{d8 selected} = (1/3)*(1/4) + (1/3)*(1/6) + (1/3)*(1/8) = (1/3)*(13/24) = 13/72 ≈ 0.1806

 

Pr{5 shows} = Pr{5 shows | d6 selected}*Pr{d6 selected} + Pr{5 shows | d8 selected}*Pr{d8 selected} = (1/3)*(1/6) + (1/3)*(1/8) = (1/3)*(7/24) = 7/72 ≈ 0.0972

 

Pr{6 shows} = Pr{6 shows | d6 selected}*Pr{d6 selected} + Pr{6 shows | d8 selected}*Pr{d8 selected} = (1/3)*(1/6) + (1/3)*(1/8) = (1/3)*(7/24) = 7/72 ≈ 0.0972

 

Pr{7 shows} = Pr{7 shows | d8 selected}*Pr{d8 selected} = (1/3)*(1/8) = 3/72 ≈ 0.0417

 

Pr{8 shows} = Pr{8 shows | d8 selected}*Pr{d8 selected} = (1/3)*(1/8) = 3/72 ≈ 0.0417

 

Sample Tables

 

Compare sample proportions (p) to probabilities (P).

 

6:30

 

Sample 1

D8

D6

D4

Joint

Face Value

n

p

Face Value

n

p

Face Value

n

p

n

p

P

1

7

0.104477612

1

13

0.2363636

1

23

0.2948718

43

0.215

0.1806

2

10

0.149253731

2

13

0.2363636

2

17

0.2179487

40

0.2

0.1806

3

9

0.134328358

3

8

0.1454545

3

15

0.1923077

32

0.16

0.1806

4

5

0.074626866

4

6

0.1090909

4

23

0.2948718

34

0.17

0.1806

5

4

0.059701493

5

12

0.2181818

 

 

16

0.08

0.0972

6

11

0.164179104

6

3

0.0545455

 

 

14

0.07

0.0972

7

13

0.194029851

 

 

 

 

 

13

0.065

0.0417

8

8

0.119402985

 

 

 

 

 

8

0.04

0.0417

Total

67

1

Total

55

1

Total

78

1

200

1

1

Sample 2

D8

D6

D4

Joint

Face Value

n

p

Face Value

n

p

Face Value

n

p

n

p

P

1

7

0.112903226

1

6

0.0895522

1

9

0.1267606

22

0.11

0.1806

2

8

0.129032258

2

8

0.119403

2

24

0.3380282

40

0.2

0.1806

3

5

0.080645161

3

19

0.2835821

3

18

0.2535211

42

0.21

0.1806

4

6

0.096774194

4

12

0.1791045

4

20

0.2816901

38

0.19

0.1806

5

6

0.096774194

5

11

0.1641791

 

 

17

0.085

0.0972

6

11

0.177419355

6

11

0.1641791

 

 

22

0.11

0.0972

7

8

0.129032258

 

 

 

 

 

8

0.04

0.0417

8

11

0.177419355

 

 

 

 

 

11

0.055

0.0417

Total

62

1

Total

67

1

Total

71

1

200

1

1

Sample 3

D8

D6

D4

Joint

Face Value

n

p

Face Value

n

p

Face Value

n

p

n

p

P

1

4

0.06779661

1

14

0.1891892

1

17

0.2833333

35

0.1813472

0.1806

2

3

0.050847458

2

12

0.1621622

2

21

0.35

36

0.1865285

0.1806

3

7

0.118644068

3

11

0.1486486

3

8

0.1333333

26

0.134715

0.1806

4

7

0.118644068

4

13

0.1756757

4

14

0.2333333

34

0.1761658

0.1806

5

7

0.118644068

5

11

0.1486486

 

 

18

0.0932642

0.0972

6

7

0.118644068

6

13

0.1756757

 

 

20

0.1036269

0.0972

7

13

0.220338983

 

 

 

 

 

13

0.0673575

0.0417

8

11

0.186440678

 

 

 

 

 

11

0.0569948

0.0417

Total

59

1

Total

74

1

Total

60

1

193

1

1

Sample 4

D8

D6

D4

Joint

Face Value

n

p

Face Value

n

p

Face Value

n

p

n

p

P

1

9

0.140625

1

9

0.1384615

1

15

0.2112676

33

0.165

0.1806

2

6

0.09375

2

12

0.1846154

2

21

0.2957746

39

0.195

0.1806

3

7

0.109375

3

7

0.1076923

3

24

0.3380282

38

0.19

0.1806

4

6

0.09375

4

14

0.2153846

4

11

0.1549296

31

0.155

0.1806

5

7

0.109375

5

13

0.2

 

 

20

0.1

0.0972

6

10

0.15625

6

10

0.1538462

 

 

20

0.1

0.0972

7

10

0.15625

 

 

 

 

 

10

0.05

0.0417

8

9

0.140625

 

 

 

 

 

9

0.045

0.0417

Total

64

1

Total

65

1

Total

71

1

200

1

1

Sample 5

D8

D6

D4

Joint

Face Value

n

p

Face Value

n

p

Face Value

n

p

n

p

P

1

5

0.080645161

1

8

0.1269841

1

22

0.2933333

35

0.175

0.1806

2

9

0.14516129

2

9

0.1428571

2

17

0.2266667

35

0.175

0.1806

3

13

0.209677419

3

13

0.2063492

3

10

0.1333333

36

0.18

0.1806

4

10

0.161290323

4

9

0.1428571

4

26

0.3466667

45

0.225

0.1806

5

7

0.112903226

5

11

0.1746032

 

 

18

0.09

0.0972

6

2

0.032258065

6

13

0.2063492

 

 

15

0.075

0.0972

7

9

0.14516129

 

 

 

 

 

9

0.045

0.0417

8

7

0.112903226

 

 

 

 

 

7

0.035

0.0417

Total

62

1

Total

63

1

Total

75

1

200

1

1

Sample 6

D8

D6

D4

Joint

Face Value

n

p

Face Value

n

p

Face Value

n

p

n

p

P

1

3

0.046875

1

14

0.2413793

1

21

0.2692308

38

0.19

0.1806

2

10

0.15625

2

13

0.2241379

2

18

0.2307692

41

0.205

0.1806

3

14

0.21875

3

9

0.1551724

3

21

0.2692308

44

0.22

0.1806

4

14

0.21875

4

6

0.1034483

4

18

0.2307692

38

0.19

0.1806

5

9

0.140625

5

7

0.1206897

 

 

16

0.08

0.0972

6

7

0.109375

6

9

0.1551724

 

 

16

0.08

0.0972

7

3

0.053571429

 

 

 

 

 

3

0.015

0.0417

8

4

0.0625

 

 

 

 

 

4

0.02

0.0417

Total

64

1.006696429

Total

58

1

Total

78

1

200

1

1

Pooled

D8

D6

D4

Joint

Face Value

n

p

Face Value

n

p

Face Value

n

p

n

p

P

1

35

0.092592593

1

64

0.1675393

1

107

0.2471132

206

0.1726739

0.1806

2

46

0.121693122

2

67

0.1753927

2

118

0.2725173

231

0.1936295

0.1806

3

55

0.145502646

3

67

0.1753927

3

96

0.221709

218

0.1827326

0.1806

4

48

0.126984127

4

60

0.1570681

4

112

0.2586605

220

0.1844091

0.1806

5

40

0.105820106

5

65

0.1701571

 

 

105

0.0880134

0.0972

6

48

0.126984127

6

59

0.1544503

 

 

107

0.0896899

0.0972

7

56

0.148148148

 

 

 

 

 

56

0.0469405

0.0417

8

50

0.132275132

 

 

 

 

 

50

0.0419111

0.0417

Total

378

1

Total

382

1

Total

433

1

1193

1

1

 

8:00

 

Sample 1

D8

D6

D4

Joint

Face Value

n

p

Face Value

n

p

Face Value

n

p

n

p

P

1

6

0.08108108

1

12

0.20689655

1

19

0.26760563

37

0.18226601

0.1806

2

9

0.12162162

2

8

0.13793103

2

19

0.26760563

36

0.1773399

0.1806

3

11

0.14864865

3

8

0.13793103

3

15

0.21126761

34

0.16748768

0.1806

4

10

0.13513514

4

14

0.24137931

4

18

0.25352113

42

0.20689655

0.1806

5

12

0.16216216

5

7

0.12068966

 

 

19

0.09359606

0.0972

6

10

0.13513514

6

9

0.15517241

 

 

19

0.09359606

0.0972

7

11

0.14864865

 

 

 

 

 

11

0.05418719

0.0417

8

5

0.06756757

 

 

 

 

 

5

0.02463054

0.0417

Total

74

1

Total

58

1

Total

71

1

203

1

1

Sample 2

D8

D6

D4

Joint

Face Value

n

p

Face Value

n

p

Face Value

n

p

n

p

P

1

5

0.06944444

1

6

0.10169492

1

21

0.30434783

32

0.16

0.1806

2

15

0.20833333

2

9

0.15254237

2

16

0.23188406

40

0.2

0.1806

3

4

0.05555556

3

10

0.16949153

3

15

0.2173913

29

0.145

0.1806

4

5

0.06944444

4

8

0.13559322

4

17

0.24637681

30

0.15

0.1806

5

15

0.20833333

5

11

0.18644068

 

 

26

0.13

0.0972

6

8

0.11111111

6

15

0.25423729

 

 

23

0.115

0.0972

7

13

0.18055556

 

 

 

 

 

13

0.065

0.0417

8

7

0.09722222

 

 

 

 

 

7

0.035

0.0417

Total

72

1

Total

59

1

Total

69

1

200

1

1

Sample 3

D8

D6

D4

Joint

Face Value

n

p

Face Value

n

p

Face Value

n

p

n

p

P

1

11

0.10784314

1

15

0.25862069

1

9

0.225

35

0.175

0.1806

2

15

0.14705882

2

10

0.17241379

2

12

0.3

37

0.185

0.1806

3

22

0.21568627

3

8

0.13793103

3

8

0.2

38

0.19

0.1806

4

15

0.14705882

4

10

0.17241379

4

11

0.275

36

0.18

0.1806

5

10

0.09803922

5

4

0.06896552

 

 

14

0.07

0.0972

6

10

0.09803922

6

11

0.18965517

 

 

21

0.105

0.0972

7

12

0.11764706

 

 

 

 

 

12

0.06

0.0417

8

7

0.06862745

 

 

 

 

 

7

0.035

0.0417

Total

102

1

Total

58

1

Total

40

1

200

1

1

Sample 4

D8

D6

D4

Joint

Face Value

n

p

Face Value

n

p

Face Value

n

p

n

p

P

1

7

0.12068966

1

11

0.14864865

1

18

0.26086957

36

0.17910448

0.1806

2

11

0.18965517

2

12

0.16216216

2

14

0.20289855

37

0.1840796

0.1806

3

6

0.10344828

3

17

0.22972973

3

18

0.26086957

41

0.2039801

0.1806

4

6

0.10344828

4

13

0.17567568

4

19

0.27536232

38

0.18905473

0.1806

5

10

0.17241379

5

9

0.12162162

 

 

19

0.09452736

0.0972

6

10

0.17241379

6

12

0.16216216

 

 

22

0.10945274

0.0972

7

4

0.06896552

 

 

 

 

 

4

0.0199005

0.0417

8

4

0.06896552

 

 

 

 

 

4

0.0199005

0.0417

Total

58

1

Total

74

1

Total

69

1

201

1

1

Sample 5

D8

D6

D4

Joint

Face Value

n

p

Face Value

n

p

Face Value

n

p

n

p

P

1

7

0.11111111

1

7

0.10769231

1

20

0.26666667

34

0.16748768

0.1806

2

5

0.07936508

2

10

0.15384615

2

25

0.33333333

40

0.19704433

0.1806

3

13

0.20634921

3

13

0.2

3

13

0.17333333

39

0.19211823

0.1806

4

7

0.11111111

4

6

0.09230769

4

17

0.22666667

30

0.14778325

0.1806

5

10

0.15873016

5

19

0.29230769

 

 

29

0.14285714

0.0972

6

9

0.14285714

6

10

0.15384615

 

 

19

0.09359606

0.0972

7

5

0.07936508

 

 

 

 

 

5

0.02463054

0.0417

8

7

0.11111111

 

 

 

 

 

7

0.03448276

0.0417

Total

63

1

Total

65

1

Total

75

1

203

1

1

Sample 6

D8

D6

D4

Joint

Face Value

n

p

Face Value

n

p

Face Value

n

p

n

p

P

1

12

0.17910448

1

9

0.13636364

1

19

0.27536232

40

0.1980198

0.1806

2

10

0.14925373

2

7

0.10606061

2

20

0.28985507

37

0.18316832

0.1806

3

10

0.14925373

3

14

0.21212121

3

17

0.24637681

41

0.2029703

0.1806

4

4

0.05970149

4

10

0.15151515

4

13

0.1884058

27

0.13366337

0.1806

5

4

0.05970149

5

10

0.15151515

 

 

14

0.06930693

0.0972

6

4

0.05970149

6

16

0.24242424

 

 

20

0.0990099

0.0972

7

13

0.22413793

 

 

 

 

 

13

0.06435644

0.0417

8

10

0.14925373

 

 

 

 

 

10

0.04950495

0.0417

Total

67

1.03010808

Total

66

1

Total

69

1

202

1

1

Pooled

D8

D6

D4

Joint

Face Value

n

p

Face Value

n

p

Face Value

n

p

n

p

P

1

48

0.11009174

1

60

0.15789474

1

106

0.2697201

214

0.17700579

0.1806

2

65

0.14908257

2

56

0.14736842

2

106

0.2697201

227

0.18775848

0.1806

3

66

0.15137615

3

70

0.18421053

3

86

0.21882952

222

0.18362283

0.1806

4

47

0.10779817

4

61

0.16052632

4

95

0.24173028

203

0.16790736

0.1806

5

61

0.13990826

5

60

0.15789474

 

 

121

0.10008271

0.0972

6

51

0.11697248

6

73

0.19210526

 

 

124

0.1025641

0.0972

7

58

0.13302752

 

 

 

 

 

58

0.04797353

0.0417

8

40

0.09174312

 

 

 

 

 

40

0.03308519

0.0417

Total

436

1

Total

380

1

Total

393

1

1209

1

1

 

Conditional Probability

 

Conditional = Joint / Prior

 

Pr{A|B} = Pr{A∩B} / Pr{B}

 

How much of B is tied up in A ?

 

Case Study 1.11

Conditional Probability

Case Study Description: Compute conditional probabilities associated with the color sequence experiment.

Suppose that we have a special box - each time we press a button on the box, it prints out a sequence of colors, in order - it prints four colors at a time. Suppose the box follows the following Probabilities for each Color Sequence:

 

 

 

 

Color Sequence

Probability CS Prints Out

BBBB

.10 = 10%

BGGB

.25 = 25%

RGGR

.05 = 05%

YYYY

.30 = 30%

BYRG

.15 = 15%

RYYB

.15 = 15%

Total

1.00 = 100%

 

Let's define the experiment: We push the button, and then the box prints out exactly one (1) of the above listed color sequences. We then note the resulting (printed out) color sequence.

Compute Pr{ blue shows 1st | blue shows 4th };

Pr{ B 1st and B 4th } = Pr{ exactly one of BBBB, BGGB shows } = Pr{ BBBB} + Pr{BGGB} =.10 + .25 = .35

 

Pr{ B 4th } = Pr{ exactly one of BBBB, BGGB, RYYB shows } = Pr{BBBB} + Pr{BGGB} +

Pr{RYYB} = .10+.25+.15 = .50

 

So, Pr{ B 1st | B 4th } = .35/ .50= .70

Compute Pr{ green shows 2nd or 3rd | yellow shows };

Pr{ G 2nd or 3rd and Y shows } = 0, since no sequences meet this requirement

 

Pr{ Y shows } = Pr{ exactly one of YYYY, BYRG, RYYB shows } = Pr{YYYY}+ Pr{BYRG}+ Pr{RYYB} = .30+.15+.15 = .60

 

So, Pr{ G 2nd or 3rd | Y shows } = 0 / .60= 0

Compute Pr{ yellow shows | red shows }.

Pr{ Y and R show } = Pr{ exactly one of BYRG, RYYB shows } = Pr{BYRG}+ Pr{RYYB } = .15 + .15 = .30

 

Pr{ R shows } = Pr{ exactly one of RGGR, BYRG, RYYB shows } = Pr{RGGR}+ Pr{BYRG}+

Pr{RYYB } = .05+.15+.15 = .35

 

So, Pr{ Y shows | R shows } = .30/.35 = 6/7 = .8571

 

Case Study 1.12

Conditional Probability II: Pair of Dice

Case Description: Compute conditional probabilities.

Suppose we have a pair of fair dice: d4(faces 1,2,3,4), d6(faces 1,2,3,4,5,6). In our experiment, we toss this pair of dice, and note the face value from each die. For simplicity, we write the outcome as (d4 result, d6 result). Assume that the dice operate independently and separately.

Case Objectives:

Identify the simple (basic) events. Compute (and justify) a probability for each simple event. 

As before, Pr{ ( any d4 face, any d6 face) } = Pr{ any d4 face }*Pr{ any d6 face } = (1/4)*(1/6) = 1/24

We have 24 equally likely pairs.

 

1

2

3

4

1

(1,1)

(2,1)

(3,1)

(4,1)

2

(1,2)

(2,2)

(3,2)

(4,2)

3

(1,3)

(2,3)

(3,3)

(4,3)

4

(1,4)

(2,4)

(3,4)

(4,4)

5

(1,5)

(2,5)

(3,5)

(4,5)

6

(1,6)

(2,6)

(3,6)

(4,6)

Suppose we observe the sum of the faces in the pair of dice. Identify the possible values of this sum, and compute (and justify) a probability for each value.

Now for the sums:

 

1

2

3

4

1

(1,1) @ 2

(2,1) @ 3

(3,1) @ 4

(4,1) @ 5

2

(1,2) @ 3

(2,2) @ 4

(3,2) @ 5

(4,2) @ 6

3

(1,3) @ 4

(2,3) @ 5

(3,3) @ 6

(4,3) @ 7

4

(1,4) @ 5

(2,4) @ 6

(3,4) @ 7

(4,4) @ 8

5

(1,5) @ 6

(2,5) @ 7

(3,5) @ 8

(4,5) @ 9

6

(1,6) @ 7

(2,6) @ 8

(3,6) @ 9

(4,6) @ 10

 

Compute the conditional probability Pr{Sum is Even|d4 shows Even}.

Pr{ Sum is Even and d4 shows Even } = Pr{ exactly one of (2,2), (2,4), (2,6), (4,2), (4,4), (4,6) shows } = 6/24 = 1/4 = .25

Pr{ d4 shows Even } =

Pr{ exactly one of (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6) shows } = 12/24 = 1/2 = .50

So, Pr{ Sum is Even | d4 shows Even } = .25 / .50 = .50

Continuing,…

 

1

2

3

4

1

(1,1) @ 2

(2,1) @ 3

(3,1) @ 4

(4,1) @ 5

2

(1,2) @ 3

(2,2) @ 4

(3,2) @ 5

(4,2) @ 6

3

(1,3) @ 4

(2,3) @ 5

(3,3) @ 6

(4,3) @ 7

4

(1,4) @ 5

(2,4) @ 6

(3,4) @ 7

(4,4) @ 8

5

(1,5) @ 6

(2,5) @ 7

(3,5) @ 8

(4,5) @ 9

6

(1,6) @ 7

(2,6) @ 8

(3,6) @ 9

(4,6) @ 10

 

Compute the conditional probability Pr{Sum is Odd|d6 shows Odd}.

Pr{ Sum is Odd and d6 shows Odd } = Pr{ exactly one of (2,1), (2,3), (2,5), (4,1), (4,3), (4,5) shows } = 6/24 = 1/4 = .25

Pr{ d6 shows Odd } =

Pr{ exactly one of (1,1), (1,3), (1,5), (2,1), (2,3), (2,5), (3,1), (3,3), (3,5), (4,1), (4,3), (4,5) shows } = 12/24 = 1/2 = .50

So, Pr{ Sum is Odd | d6 shows Odd } = .25/.50 = 1/2 = .50

HR1 – Summer Version A, Case Three

 

Case Three | Color Slot Machine | Conditional Probabilities

 

Here is our slot machine – on each trial, it produces a 10-color sequence, using the table below:

 

Sequence*

Probability

RRBBRRYRRR

.10

RRGGRGBRRB

.10

BBYYGGYGBR

.15

GRRGGYBRGG

.10

BGYGYRYGYY

.25

RRYYGRRBBY

.10

YYGBYYBGRR

.20

Total

1.00

*B-Blue, G-Green, R-Red, Y-Yellow, Sequence is numbered from left to right: (1st 2nd 3rd 4th 5th6th7th 8th 9th 10th )

Compute the following conditional probabilities:

 

Pr{ Yellow Shows Exactly Twice | Blue Shows}

 

 

Sequence*

Probability

RRBBRRYRRR

.10

RRGGRGBRRB

.10

BBYYGGYGBR

.15

GRRGGYBRGG

.10

BGYGYRYGYY

.25

RRYYGRRBBY

.10

YYGBYYBGRR

.20

Total

1.00

 

Pr{Blue Shows} = Pr{One of RRBBRRYRRR, RRGGRGBRRB, BBYYGGYGBR, GRRGGYBRGG, BGYGYRYGYY, RRYYGRRBBY, YYGBYYBGRR Shows} =

Pr{RRBBRRYRRR} + Pr{RRGGRGBRRB} + Pr{BBYYGGYGBR} + Pr{GRRGGYBRGG} + Pr{BGYGYRYGYY} + Pr{RRYYGRRBBY} + Pr{YYGBYYBGRR} = .1+.1+.15+.1+.25+.1+.2 = 1.00

 

Sequence*

Probability

 

 

Total

0

 

 

Pr{ Yellow Shows Exactly Twice and  Blue Shows} = 0

 

 

Pr{ Yellow Shows Exactly Twice | Blue Shows} = Pr{ Yellow Shows Exactly Twice and Blue Shows}/Pr{Blue Shows} = 0/1 =0

 

Pr{ Green Shows | “BR” Shows }

 

Sequence*

Probability

RRBBRRYRRR

.10

RRGGRGBRRB

.10

BBYYGGYGBR

.15

GRRGGYBRGG

.10

Total

.45

 

Pr{ “BR” Shows } = Pr{One of RRBBRRYRRR, RRGGRGBRRB, BBYYGGYGBR, GRRGGYBRGG Shows} = Pr{RRBBRRYRRR}+ Pr{ RRGGRGBRRB}+

Pr{BBYYGGYGBR}+ Pr{GRRGGYBRGG} = .1+.1+.15+.1 = .45

 

 

Pr{Green Shows and “BR” Shows}

 

Sequence*

Probability

RRGGRGBRRB

.10

BBYYGGYGBR

.15

GRRGGYBRGG

.10

Total

.35

 

Pr{ Green Shows and “BR” Shows } = Pr{One of RRGGRGBRRB, BBYYGGYGBR, GRRGGYBRGG Shows} = Pr{ RRGGRGBRRB}+

Pr{BBYYGGYGBR}+ Pr{GRRGGYBRGG} =.1+.15+.1 = .35

 

Pr{ Green Shows | “BR” Shows } = Pr{ Green Shows and “BR” Shows }/Pr{ “BR” Shows } = .35/.45 = 7/9

 

 

Pr{ Red Shows | Green Shows}

 

Sequence*

Probability

RRGGRGBRRB

.10

BBYYGGYGBR

.15

GRRGGYBRGG

.10

BGYGYRYGYY

.25

RRYYGRRBBY

.10

YYGBYYBGRR

.20

Total

.90

 

Pr{Green Shows} = Pr{One of RRGGRGBRRB, BBYYGGYGBR, GRRGGYBRGG, BGYGYRYGYY, RRYYGRRBBY, YYGBYYBGRR Shows} =

Pr{RRBBRRYRRR} + Pr{RRGGRGBRRB} + Pr{BBYYGGYGBR} + Pr{GRRGGYBRGG} + Pr{BGYGYRYGYY} + Pr{RRYYGRRBBY} + Pr{YYGBYYBGRR} = .1+.15+.1+.25+.1+.2 = .90

 

Pr{ Red Shows and Green Shows}

 

Sequence*

Probability

RRGGRGBRRB

.10

BBYYGGYGBR

.15

GRRGGYBRGG

.10

BGYGYRYGYY

.25

RRYYGRRBBY

.10

YYGBYYBGRR

.20

Total

.90

 

Pr{ Red and Green Show } = Pr{One of RRGGRGBRRB, BBYYGGYGBR, GRRGGYBRGG, BGYGYRYGYY, RRYYGRRBBY, YYGBYYBGRR Shows} =

Pr{RRBBRRYRRR} + Pr{RRGGRGBRRB} + Pr{BBYYGGYGBR} + Pr{GRRGGYBRGG} + Pr{BGYGYRYGYY} + Pr{RRYYGRRBBY} + Pr{YYGBYYBGRR} = .1+.15+.1+.25+.1+.2 = .90

 

Pr{ Red Shows | Green Shows} = Pr{ Red Shows and Green Shows}/Pr{ Green Shows} = .90/.90 = 1

 

HR1 – Spring 2008, Case Four

 

Case Four: Color Slot Machine, Computation of Conditional Probabilities

 

Here is our slot machine – on each trial, it produces a 10-color sequence, using the table below:

 

Sequence*

Probability

RRBBR RYRRB

.10

RRGGRGBRRB

.10

BBYYRGYGBR

.15

GRRGRGBRGB

.10

BGYGYRYGYY

.25

RRGYGRRBBB

.10

YYGBYYBGRR

.20

Total

1.00

*B-Blue, G-Green, R-Red, Y-Yellow, Sequence is numbered as 1st to 6th , from left to right: (1st 2nd 3rd 4th 5th6th7th 8th 9th 10th )

Compute the following conditional probabilities:

 

1. Pr{Red Shows Somewhere in the 1st ─ 4th slots | Yellow Shows Somewhere in the 7th ─ 10th slots}

 

Pr{Red Shows in the 1st – 4th slots|Yellow Shows in the 7th – 10th slots} =

 

Pr{Red Shows in the 1st – 4th slots and Yellow Shows in the 7th – 10th slots}/ Pr{ Yellow Shows in the 7th – 10th slots}

 

Sequence*

Probability

RRBBR RYRRB

.10

BBYYRGYGBR

.15

BGYGYRYGYY

.25

Total

0.50

 

Pr{ Yellow Shows in the 7th – 10th slots} = Pr{One of RRBBRRYRRB, BBYYRGYGBR, BGYGYRYGYY shows} =

Pr{RRBBRRYRRB}+ Pr{BBYYRGYGBR}+ Pr{BGYGYRYGYY} = .10+.15+.25 = .50

 

Sequence*

Probability

RRBBR RYRRB

.10

Total

0.10

 

Pr{ Red Shows in the 1st – 4th slots and Yellow Shows in the 7th – 10th slots } = Pr{One of RRBBRRYRRB shows} = .10

 

Pr{Red Shows in the 1st – 4th slots|Yellow Shows in the 7th – 10th slots} = .10/.50 = .20

 

2. Pr{Green Shows Anywhere  | “RB” Shows Anywhere}

 

Pr{Green Shows Anywhere|”RB” Shows Anywhere} =

 

Pr{ Green Shows Anywhere and ”RB” Shows Anywhere }/ Pr{”RB” Shows Anywhere}

 

Sequence*

Probability

RRBBR RYRRB

.10

RRGGRGBRRB

.10

RRGYGRRBBB

.10

Total

0.30

Pr{”RB” Shows Anywhere} = Pr{One of  RRBBRRYRRB, RRGGRGBRRB, RRGYGRRBBB Shows} = Pr{RRBBRRYRRB}+Pr{RRGGRGBRRB}+Pr{RRGYGRRBBB} =.10+.10+.10 = .30

 

Sequence*

Probability

RRGGRGBRRB

.10

RRGYGRRBBB

.10

Total

0.20

 

Pr{ Green Shows Anywhere and ”RB” Shows Anywhere } = Pr{One of  RRGGRGBRRB, RRGYGRRBBB Shows} = Pr{RRGGRGBRRB}+Pr{RRGYGRRBBB} =.10+.10 = .20

 

Pr{Green Shows Anywhere|”RB” Shows Anywhere} = .20/.30

 

3. Pr{Yellow Shows Anywhere | Blue Shows Anywhere}

 

Pr{Yellow Shows Anywhere | Blue Shows Anywhere} =

Pr{Yellow Shows Anywhere and Blue Shows Anywhere}/Pr{ Blue Shows Anywhere}

 

Sequence*

Probability

RRBBR RYRRB

.10

RRGGRGBRRB

.10

BBYYRGYGBR

.15

GRRGRGBRGB

.10

BGYGYRYGYY

.25

RRGYGRRBBB

.10

YYGBYYBGRR

.20

Total

1.00

Pr{ Blue Shows Anywhere} = Pr{one of RRBBRRYRRB, RRGGRGBRRB, BBYYRGYGBR, GRRGRGBRGB, BGYGYRYGYY, RRGYGRRBBB, YYGBYYBGRR Shows} =Pr{RRBBRRYRRB}+Pr{RRGGRGBRRB}+Pr{ BBYYRGYGBR}+Pr{GRRGRGBRGB}+Pr{BGYGYRYGYY}+Pr{RRGYGRRBBB}+Pr{YYGBYYBGRR} = .10+.10+.15+.10+.25+.10+.20 = 1.00

 

Sequence*

Probability

RRBBR RYRRB

.10

BBYYRGYGBR

.15

BGYGYRYGYY

.25

RRGYGRRBBB

.10

YYGBYYBGRR

.20

Total

0.80

 

 

Pr{Yellow Shows Anywhere and Blue Shows Anywhere} = Pr{one of RRBBRRYRRB, BBYYRGYGBR, BGYGYRYGYY, RRGYGRRBBB, YYGBYYBGRR Shows} =Pr{RRBBRRYRRB}+ Pr{BBYYRGYGBR}+Pr{BGYGYRYGYY}+Pr{RRGYGRRBBB}+Pr{YYGBYYBGRR} = .10+.15+.25+.10+.20 = .80

 

Pr{Yellow Shows Anywhere | Blue Shows Anywhere} = .80/1.00 = .80

 

 

Case Study 1.13

Conditional Probability

Case Description: Compute conditional probabilities for pairs of draws (without replacement).

Here is our bowl, in tabular form:

Color

# in Bowl

Proportion of Bowl

Blue

5

5/9

Green

3

3/9

Red

1

1/9

Total

9

1

 

Suppose that on each trial of this experiment that we make two (2) draws without replacement from the bowl.

Compute Pr{ green shows 2nd | red shows 1st };

Here is our bowl, after "red shows 1st", in tabular form:

 

Color

# in Bowl – Before 1st Draw

# in Bowl – After 1st Draw

Blue

5

5 – 0 = 5

Green

3

3 – 0 = 3

Red

1

1 – 1 = 0

Total

9

8

With the red chip out of the bowl, 3 of the 8 surviving chips are green. So, Pr{G 2nd | R 1st} = (3-0) / (9-1) = 3/8

Compute Pr{ red shows 2nd | red shows 1st };

Pr{R 2nd | R 1st} = (1-1) / (9-1) = 0/8 = 0. There are 1-1=0 surviving red chips after the first draw.

Compute Pr{ blue shows 2nd | blue shows 1st }.

Here is our bowl, after "blue shows 1st", in tabular form:

Color

# in Bowl – Before 1st Draw

# in Bowl – After 1st Draw

Blue

5

5 – 1 = 4

Green

3

3 – 0 = 3

Red

1

1 – 0 = 1

Total

9

8

Pr{B 2nd | B 1st} = (5-1)/(9-1) = 4/8. After the first draw, 4 of 8 surviving chips are blue.

HR1 – Fall 2004, Case Three

Case Three

Conditional Probability

Color Bowl/Draws without Replacement

 

We have a bowl containing the following colors and counts of balls (color@count):

 

Blue @ 5, Green @ 1, Red @ 2, Yellow @ 3

 

Each trial of our experiment consists of three (3) draws without replacement from the bowl.

 

Compute these directly.

 

Color

Count

B

5

G

1

R

2

Y

3

Total

11

 

Pr{ green shows 2nd | green shows 1st}

 

Color

Count

B

5

G

1

R

2

Y

3

Total

11

ß green shows 1st

Color

Count

B

5

G

0

R

2

Y

3

Total

10

 

Pr{ green shows 2nd  | green shows 1st} = 0/10

  

Pr{ yellow shows 3rd | green shows 1st, blue shows 2nd}

 

Color

Count

B

5

G

1

R

2

Y

3

Total

11

ß green shows 1st

Color

Count

B

5

G

0

R

2

Y

3

Total

10

ß blue shows 2nd

Color

Count

B

4

G

0

R

2

Y

3

Total

9

 

Pr{ yellow shows 3rd | green shows 1st, blue shows 2nd} = 3/9

 

Pr{ red shows 3rd | green shows 1st, red shows 2nd }

 

Color

Count

B

5

G

1

R

2

Y

3

Total

11

ß green shows 1st

Color

Count

B

5

G

0

R

2

Y

3

Total

10

ß red shows 2nd

Color

Count

B

5

G

0

R

1

Y

3

Total

9

 

Pr{ red shows 3rd | green shows 1st, red shows 2nd } = 1/9