Thermodynamics and Statistical Mechanics

Thermodynamics and Statistical MechanicsProbabilities

Thermo & Stat Mech - Spring 2006 Class 16

Pair of DiceFor one die, the probability of any face coming up is the same, 1/6. Therefore, it is equally probable that any number from one to six will come up.

For two dice, what is the probability that the total will come up 2, 3, 4, etc up to 12?


ProbabilityTo calculate the probability of a particular outcome, count the number of all possible results. Then count the number that give the desired outcome. The probability of the desired outcome is equal to the number that gives the desired outcome divided by the total number of outcomes. Hence, 1/6 for one die.


Pair of DiceList all possible outcomes (36) for a pair of dice.Total CombinationsHow Many21+1131+2, 2+1241+3, 3+1, 2+2351+4, 4+1, 2+3, 3+2461+5, 5+1, 2+4, 4+2, 3+35


Pair of DiceTotal CombinationsHow Many71+6, 6+1, 2+5, 5+2, 3+4, 4+3682+6, 6+2, 3+5, 5+3, 4+4593+6, 6+3, 4+5, 5+44104+6, 6+4, 5+53115+6, 6+52126+61 Sum = 36


Probabilities for Two Dice


Probabilities for Two Dice


Chart1

0.0277777778

0.0555555556

0.0833333333

0.1111111111

0.1388888889

0.1666666667

0.1388888889

0.1111111111

0.0833333333

0.0555555556

0.0277777778

Number

Probability

Dice

Sheet1

Distributions

Dice Pair

nprob

20.0277777778

30.0555555556

40.0833333333

50.1111111111

60.1388888889

70.1666666667

80.1388888889

90.1111111111

100.0833333333

110.0555555556

120.0277777778

Binomial Distribution

p =0.5

q =0.5

N =6120

npCalcExp

00.0156251.8753110100026.55187513.05

10.0937511.25143312231454.6087511.0712142643

20.23437528.125211366324219.9631251

30.312537.53846821261140.023752

40.23437528.1253465563913635.721252

50.0937511.2595202004536.9056253

60.0156251.875100010069.1506253

3.32.852.8532.83.05Mean =2.975pN =33

1.51986841541.35185058350.85293610551.51657508880.81240384051.0712142643SD =1.23465582251.22474487143

0.33985290940.302282980.19072231120.33911649920.18165902120.23953079133

0.17017148213

4

4

4

4

4

4

4

4

4

&A

Page &P

Sheet1

&A

Page &P

Number

Probability

Dice

Sheet2

0

0

0

0

0

0

0

&A

Page &P

Successes

Probabilty


Sheet3

00

00

00

00

00

00

00

&A

Page &P

Calc

Exp

No of Heads

No of occurences

Experimental Coin Toss

Sheet4


p =0.5Sqrt(Npq) =2.23606797752.5066282746

q =0.5Mean =Np =105.6049912164

N =200.17841241160.1784124116

nBinomialGaussian

00.00000095370.0000080999

10.00001907350.0000541551

20.00018119810.0002964424

30.00108718870.0013285628

40.00462055210.0048748912

50.01478576660.0146449826

60.03696441650.0360208447

70.0739288330.0725370735

80.12013435360.119593416

90.16017913820.1614342259

100.1761970520.1784124116

110.16017913820.1614342259

120.12013435360.119593416

130.0739288330.0725370735

140.03696441650.0360208447

150.01478576660.0146449826

160.00462055210.0048748912

170.00108718870.0013285628

180.00018119810.0002964424

190.00001907350.0000541551

200.00000095370.0000080999

&A

Page &P

Sheet4

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

&A

Page &P

Binomial

Gaussian

No. of Heads

Probability

Twenty Coins

Sheet5

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&A

Page &P

No. of Heads

Probability

Gaussian

Sheet6

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&A

Page &P

No. of Heads

Probability

Binomial

Sheet7

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&A

Page &P

No. of Heads

Probability

Gaussian

Sheet8

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

&A

Page &P

Binomial

Gaussian

No. of Heads

Probability

Twenty Coins

Sheet9

&A

Page &P

Sheet10

&A

Page &P

Sheet11

&A

Page &P

Sheet12

&A

Page &P

Sheet13

&A

Page &P

Sheet14

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Page &P

Sheet15

&A

Page &P

Sheet16

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Microstates and MacrostatesEach possible outcome is called a microstate.The combination of all microstates that give the same number of spots is called a macrostate.The macrostate that contains the most microstates is the most probable to occur.


Combining ProbabilitiesIf a given outcome can be reached in two (or more) mutually exclusive ways whose probabilities are pA and pB, then the probability of that outcome is: pA + pB.

This is the probability of having either A or B.


Combining ProbabilitiesIf a given outcome represents the combination of two independent events, whose individual probabilities are pA and pB, then the probability of that outcome is: pA pB.

This is the probability of having both A and B.


ExamplePaint two faces of a die red. When the die is thrown, what is the probability of a red face coming up?


Another ExampleThrow two normal dice. What is the probability of two sixes coming up?


Complicationsp is the probability of success. (1/6 for one die)q is the probability of failure. (5/6 for one die)

p + q = 1, or q = 1 p

When two dice are thrown, what is the probability of getting only one six?


ComplicationsProbability of the six on the first die and not the second is:

Probability of the six on the second die and not the first is the same, so:


SimplificationProbability of no sixes coming up is:

The sum of all three probabilities is:p(2) + p(1) + p(0) = 1


Simplificationp(2) + p(1) + p(0) = 1p + 2pq + q =1(p + q) = 1

The exponent is the number of dice (or tries).Is this general?


Three Dice(p + q) = 1p + 3pq + 3pq + q = 1p(3) + p(2) + p(1) + p(0) = 1

It works! It must be general!(p + q)N = 1


Binomial DistributionProbability of n successes in N attempts(p + q)N = 1

where, q = 1 p.


Thermodynamic ProbabilityThe term with all the factorials in the previous equation is the number of microstates that will lead to the particular macrostate. It is called the thermodynamic probability, wn.


MicrostatesThe total number of microstates is:

For a very large number of particles


Mean of Binomial Distribution


Standard Deviation (s)


Standard Deviation


For a Binomial Distribution


CoinsToss 6 coins. Probability of n heads:


For Six Coins


Chart2

0.015625

0.09375

0.234375

0.3125

0.234375

0.09375

0.015625

Successes

Probabilty


Sheet1

Distributions

Dice Pair

nprob

20.0277777778

30.0555555556

40.0833333333

50.1111111111

60.1388888889

70.1666666667

80.1388888889

90.1111111111

100.0833333333

110.0555555556

120.0277777778


p =0.5

q =0.5

N =6150

npCalcExp

00.0156252.343753100101029.328133333313.05

10.0937514.06259114201940.704411.0712142643

20.23437535.15625356956547044.42822222221

30.312546.875448876961320.70595555562

40.23437535.156254184686916431.27115555562

50.0937514.0625141331247049.13128888893

60.0156252.3437540001302433.02417777783

702.962.962.9633.563.32Mean =3.1266666667pN =33

801.11283421951.07628992381.24835892271.32664991611.23547561691.2560254774SD =1.23448594791.22474487143

900.22256684390.21525798480.24967178450.26532998320.24709512340.25120509553

1000.23257018063

1104

1204

1304

1404

1504

1604

1704

1804

1904

200

210

220

230

240

250

260

270

280

290

300

310

320

330

340

350

360

370

380

390

400

410

420

430

440

450

460

470

480

490

500

510

520

530

540

550

560

570

580

590

600

610

620

630

640

650

660

670

680

690

700

710

720

730

740

750

760

770

780

790

800

810

820

830

840

850

860

870

880

890

900

910

920

930

940

950

960

970

980

990

1000

&A

Page &P

Sheet1

&A

Page &P

Number

Probability

Dice

Sheet2

&A

Page &P

Successes

Probabilty


Sheet3

&A

Page &P

Calc

Exp

No of Heads

No of occurences


Sheet4


p =0.1666666667Sqrt(Npq) =1.66666666672.5066282746

q =0.8333333333Mean =Np =3.33333333334.1777137911

N =200.23936536820.2393653682

nBinomialGaussian

00.02608405330.0323945799

10.10433621320.0898364794

20.19823880510.1738149317

30.23788656610.2346256164

40.20220358120.2209620842

50.1294102920.1451824347

60.0647051460.0665525008

70.02588205840.0212847557

80.0084116690.0047492709

90.00224311170.0007393315

100.00049348460.0000802981

110.00008972450.0000060845

120.00001345870.0000003217

130.00000165650.0000000119

140.00000016560.0000000003

150.00000001330

160.00000000080

1705.99902724909831E-16

1803.65745487773748E-18

1901.55571882066023E-20

202.73511122779126E-164.61675917602385E-23

&A

Page &P

Sheet4

&A

Page &P

Binomial

Gaussian

No. of Heads

Probability

Twenty Coins

Sheet5

&A

Page &P

No. of Heads

Probability

Gaussian

Sheet6

&A

Page &P

No. of Heads

Probability

Binomial

Sheet7

&A

Page &P

No. of Heads

Probability

Gaussian

Sheet8

&A

Page &P

Binomial

Gaussian

No. of Heads

Probability

Twenty Coins

Sheet9

&A

Page &P

Sheet10

&A

Page &P

Sheet11

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Page &P

Sheet12

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Page &P

Sheet13

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Page &P

Sheet14

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Page &P

Sheet15

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Page &P

Sheet16

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Page &P

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Page &P

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Page &P

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Page &P

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Page &P

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Page &P

For 100 Coins


Chart3

7.8886090522101E-31

7.88860905221012E-29

3.90486148084402E-27

1.27558808374238E-25

3.09330110307528E-24

5.93913811790455E-23

9.40363535334876E-22

1.26277389030683E-20

1.4679746474817E-19

1.50059630631463E-18

1.36554263874631E-17

1.11726215897426E-16

8.28636101239238E-16

0

0

0

0

0

0

0.0000000001

0.0000000004

0.0000000016

0.0000000058

0.0000000196

0.0000000629

0.0000001913

0.0000005519

0.0000015125

0.0000039434

0.0000097904

0.0000231707

0.0000523209

0.000112817

0.0002324713

0.0004581053

0.0008638557

0.0015597394

0.0026979276

0.00447288

0.0071107323

0.0108438667

0.0158690732

0.0222922695

0.0300686426

0.0389525598

0.0484742966

0.0579583981

0.0665905

0.0735270104

0.0780286641

0.0795892374

0.0780286641

0.0735270104

0.0665905

0.0579583981

0.0484742966

0.0389525598

0.0300686426

0.0222922695

0.0158690732

0.0108438667

0.0071107323

0.00447288

0.0026979276

0.0015597394

0.0008638557

0.0004581053

0.0002324713

0.000112817

0.0000523209

0.0000231707

0.0000097904

0.0000039434

0.0000015125

0.0000005519

0.0000001913

0.0000000629

0.0000000196

0.0000000058

0.0000000016

0.0000000004

0.0000000001

0

0

0

0

0

0

8.28636101239238E-16

1.11726215897426E-16

1.36554263874631E-17

1.50059630631463E-18

1.4679746474817E-19

1.26277389030683E-20

9.40363535334876E-22

5.93913811790455E-23

3.09330110307528E-24

1.27558808374238E-25

3.90486148084402E-27

7.88860905221012E-29

7.8886090522101E-31

Successes

Probabilty


Sheet1

Distributions

Dice Pair

nprob

20.0277777778

30.0555555556

40.0833333333

50.1111111111

60.1388888889

70.1666666667

80.1388888889

90.1111111111

100.0833333333

110.0555555556

120.0277777778


p =0.5

q =0.5

N =100150

npCalcExp

07.8886090522101E-311.18329135783152E-283100101029.328133333313.05

17.88860905221012E-291.18329135783152E-269114201940.704411.0712142643

23.90486148084402E-275.85729222126602E-25356956547044.42822222221

31.27558808374238E-251.91338212561357E-23448876961320.70595555562

43.09330110307528E-244.63995165461292E-224184686916431.27115555562

55.93913811790455E-238.90870717685682E-21141331247049.13128888893

69.40363535334876E-221.41054530300231E-1940001302433.02417777783

71.26277389030683E-202.962.962.9633.563.32Mean =3.1266666667pN =503

81.4679746474817E-191.11283421951.07628992381.24835892271.32664991611.23547561691.2560254774SD =1.234485947953

91.50059630631463E-180.22256684390.21525798480.24967178450.26532998320.24709512340.25120509553

101.36554263874631E-170.23257018063

111.11726215897426E-164

128.28636101239238E-164

1304

1404

1504

1604

1704

1804

190.00000000014

200.0000000004

210.0000000016

220.0000000058

230.0000000196

240.0000000629

250.0000001913

260.0000005519

270.0000015125

280.0000039434

290.0000097904

300.0000231707

310.0000523209

320.000112817

330.0002324713

340.0004581053

350.0008638557

360.0015597394

370.0026979276

380.00447288

390.0071107323

400.0108438667

410.0158690732

420.0222922695

430.0300686426

440.0389525598

450.0484742966

460.0579583981

470.0665905

480.0735270104

490.0780286641

500.0795892374

510.0780286641

520.0735270104

530.0665905

540.0579583981

550.0484742966

560.0389525598

570.0300686426

580.0222922695

590.0158690732

600.0108438667

610.0071107323

620.00447288

630.0026979276

640.0015597394

650.0008638557

660.0004581053

670.0002324713

680.000112817

690.0000523209

700.0000231707

710.0000097904

720.0000039434

730.0000015125

740.0000005519

750.0000001913

760.0000000629

770.0000000196

780.0000000058

790.0000000016

800.0000000004

810.0000000001

820

830

840

850

860

870

888.28636101239238E-16

891.11726215897426E-16

901.36554263874631E-17

911.50059630631463E-18

921.4679746474817E-19

931.26277389030683E-20

949.40363535334876E-22

955.93913811790455E-23

963.09330110307528E-24

971.27558808374238E-25

983.90486148084402E-27

997.88860905221012E-29

1007.8886090522101E-31

&A

Page &P

Sheet1

&A

Page &P

Number

Probability

Dice

Sheet2

&A

Page &P

Successes

Probabilty


Sheet3

&A

Page &P

Calc

Exp

No of Heads

No of occurences


Sheet4


p =0.1666666667Sqrt(Npq) =1.66666666672.5066282746

q =0.8333333333Mean =Np =3.33333333334.1777137911

N =200.23936536820.2393653682

nBinomialGaussian

00.02608405330.0323945799

10.10433621320.0898364794

20.19823880510.1738149317

30.23788656610.2346256164

40.20220358120.2209620842

50.1294102920.1451824347

60.0647051460.0665525008

70.02588205840.0212847557

80.0084116690.0047492709

90.00224311170.0007393315

100.00049348460.0000802981

110.00008972450.0000060845

120.00001345870.0000003217

130.00000165650.0000000119

140.00000016560.0000000003

150.00000001330

160.00000000080

1705.99902724909831E-16

1803.65745487773748E-18

1901.55571882066023E-20

202.73511122779126E-164.61675917602385E-23

&A

Page &P

Sheet4

&A

Page &P

Binomial

Gaussian

No. of Heads

Probability

Twenty Coins

Sheet5

&A

Page &P

No. of Heads

Probability

Gaussian

Sheet6

&A

Page &P

No. of Heads

Probability

Binomial

Sheet7

&A

Page &P

No. of Heads

Probability

Gaussian

Sheet8

&A

Page &P

Binomial

Gaussian

No. of Heads

Probability

Twenty Coins

Sheet9

&A

Page &P

Sheet10

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Page &P

Sheet11

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Page &P

Sheet12

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Page &P

Sheet13

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Page &P

Sheet14

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Sheet15

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Page &P

Sheet16

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Page &P

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Page &P

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Page &P

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Page &P

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Page &P

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Page &P

For 1000 Coins


Chart1

9.33263618503283E-302

9.33263618503232E-299

4.66165177442391E-296

1.55077615695827E-293

3.86530957121872E-291

7.69969666586726E-289

1.2768663637564E-286

1.81315023653398E-284

2.25057273109768E-282

2.48063127694337E-280

2.45830559545074E-278

2.2124750359058E-276

1.8234481754256E-274

1.38582061332353E-272

9.77003532393038E-271

6.42216988626395E-269

3.95364833623102E-267

2.28846468403033E-265

1.24975599133427E-263

6.45926517626487E-262

3.16826956895775E-260

1.47852579884703E-258

6.57943980486894E-257

2.79769223007052E-255

1.13889387865781E-253

4.44624170228036E-252

1.66734063835505E-250

6.01477696947373E-249

2.09013499689201E-247

7.00555592061776E-246

2.26746493297316E-244

7.09497091930352E-243

2.14844588150148E-241

6.30210791907139E-240

1.79239363462991E-238

4.94700643157885E-237

1.3260725573537E-235

3.45495660888891E-234

8.75558740621108E-233

2.15971156019862E-231

5.18870702337748E-230

1.2149167664493E-228

2.77405995005938E-227

6.1803475166436E-226

1.34422558487006E-224

2.85573257585713E-223

5.92874915205155E-222

1.20340993426742E-220

2.38927014032692E-219

4.64201055834918E-218

8.82910408198064E-217

1.64463703487867E-215

3.00146258865374E-214

5.36865383781807E-213

9.41502811928519E-212

1.61938483651697E-210

2.73271191162254E-209

4.52575446416059E-208

7.35825251673049E-207

1.17482607978979E-205

1.84251890180376E-204

2.83929142245154E-203

4.30015265432604E-202

6.40244950755179E-201

9.37358623215057E-200

1.3497964174296E-198

1.91221159135873E-197

2.66568003929694E-196

3.65746981862339E-195

4.94023459559014E-194

6.57051201213456E-193

8.6064453116697E-192

1.11047051313065E-190

1.41166662491137E-189

1.76839859634159E-188

2.18338280028321E-187

2.65740669771297E-186

3.18888803725574E-185

3.77351751075244E-184

4.40402929735944E-183

5.07013872858475E-182

5.75867608678799E-181

6.45393088263153E-180

7.13820307259769E-179

7.79253835425205E-178

8.39760603822981E-177

8.93466223834863E-176

9.38653021362197E-175

9.73852509663224E-174

9.9792526832912E-173

1.01012213271975E-171

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2.18447176192991E-117

3.94132287234974E-118

7.05631421198107E-119

1.25353346589304E-119

2.20951844752017E-120

3.86406395164891E-121

6.70435480473705E-122

1.15402828606121E-122

1.97062140075961E-123

3.3380853167071E-124

5.60891815176019E-125

9.3481969195998E-126

1.54533639415977E-126

2.53363292530832E-127

4.11972833383485E-128

6.64318141998852E-129

1.06229320505037E-129

1.68442325337838E-130

2.64834176253727E-131

4.128477343447E-132

6.38080696680427E-133

9.77704293300596E-134

1.48512044551996E-134

2.23621584325406E-135

3.33763558694655E-136

4.93755723298257E-137

7.23948826829104E-138

1.05196225408811E-138

1.5148256458868E-139

2.16156627552349E-140

3.05626246482209E-141

4.28155220014963E-142

5.94254116516754E-143

8.17099410210585E-144

1.1129617392198E-144

1.50161504497918E-145

2.00668828207852E-146

2.65591096157462E-147

3.48119402873039E-148

4.51847983413112E-149

5.80729088039367E-150

7.38990844014101E-151

9.31012086946846E-152

1.1611499061922E-152

1.43351840270632E-153

1.75172091810535E-154

2.1185426557152E-155

2.53561593021856E-156

3.00307585031455E-157

3.51922951208756E-158

4.08026610097085E-159

4.68003795545682E-160

5.30994295279823E-161

5.95893598036281E-162

6.6136914321448E-163

7.25892962064714E-164

7.87790811543057E-165

8.4530651238585E-166

8.96678731370675E-167

9.40226042827903E-168

9.74434928619933E-169

9.98044585480713E-170

1.01012213271981E-170

1.01012213271975E-171

9.9792526832912E-173

9.73852509663224E-174

9.38653021362197E-175

8.93466223834863E-176

8.39760603822981E-177

7.79253835425205E-178

7.13820307259769E-179

6.45393088263153E-180

5.75867608678799E-181

5.07013872858475E-182

4.40402929735944E-183

3.77351751075244E-184

3.18888803725574E-185

2.65740669771297E-186

2.18338280028321E-187

1.76839859634159E-188

1.41166662491137E-189

1.11047051313065E-190

8.6064453116697E-192

6.57051201213456E-193

4.94023459559014E-194

3.65746981862339E-195

2.66568003929694E-196

1.91221159135873E-197

1.3497964174296E-198

9.37358623215057E-200

6.40244950755179E-201

4.30015265432604E-202

2.83929142245154E-203

1.84251890180376E-204

1.17482607978979E-205

7.35825251673049E-207

4.52575446416059E-208

2.73271191162254E-209

1.61938483651697E-210

9.41502811928519E-212

5.36865383781807E-213

3.00146258865374E-214

1.64463703487867E-215

8.82910408198064E-217

4.64201055834918E-218

2.38927014032692E-219

1.20340993426742E-220

5.92874915205155E-222

2.85573257585713E-223

1.34422558487006E-224

6.1803475166436E-226

2.77405995005938E-227

1.2149167664493E-228

5.18870702337748E-230

2.15971156019862E-231

8.75558740621108E-233

3.45495660888891E-234

1.3260725573537E-235

4.94700643157885E-237

1.79239363462991E-238

6.30210791907139E-240

2.14844588150148E-241

7.09497091930352E-243

2.26746493297316E-244

7.00555592061776E-246

2.09013499689201E-247

6.01477696947373E-249

1.66734063835505E-250

4.44624170228036E-252

1.13889387865781E-253

2.79769223007052E-255

6.57943980486894E-257

1.47852579884703E-258

3.16826956895775E-260

6.45926517626487E-262

1.24975599133427E-263

2.28846468403033E-265

3.95364833623102E-267

6.42216988626395E-269

9.77003532393038E-271

1.38582061332353E-272

1.8234481754256E-274

2.2124750359058E-276

2.45830559545074E-278

2.48063127694337E-280

2.25057273109768E-282

1.81315023653398E-284

1.2768663637564E-286

7.69969666586726E-289

3.86530957121872E-291

1.55077615695827E-293

4.66165177442391E-296

9.33263618503232E-299

9.33263618503283E-302

Successes

Probabilty


Sheet1

Distributions

Dice Pair

nprob

20.0277777778

30.0555555556

40.0833333333

50.1111111111

60.1388888889

70.1666666667

80.1388888889

90.1111111111

100.0833333333

110.0555555556

120.0277777778


p =0.5

q =0.5

N =1000275271

npCalcExp

09.33263618503283E-3022.56647495088403E-299200020000000018.17496859513.05

19.33263618503232E-2992.56647495088389E-296281214521244228113.63501487611.0712142643

24.66165177442391E-2961.28195423796658E-293633113321135610612664.84605619831

31.55077615695827E-2934.26463443163524E-2917192107526610952130.01502148762

43.86530957121872E-2911.06296013208515E-28877989611867211030874.77629090912

57.69969666586726E-2892.1174165831135E-2862642131235212130102.49277355373

61.2768663637564E-2863.51138250033009E-2844001010101002435.65175537193

71.81315023653398E-2843.62.883.362.83.162.882.83.322.82.43.16Mean =3.0145454545pN =5003

82.25057273109768E-2821.03846039891.17711511761.01508620321.46969384571.34699665921.17711511761.24579292021.22376468331.26491106410.9380831521.12SD =1.220420018615.81138830083

92.48063127694337E-2800.20769207980.23542302350.20301724060.29393876910.26939933180.23542302350.2491585840.24475293670.25298221280.18761663040.224M of Ms =3.01454545453

102.45830559545074E-278SD of Ms =0.32328888393

112.2124750359058E-2764

121.8234481754256E-2744

131.38582061332353E-2724

149.77003532393038E-2714

156.42216988626395E-2694

163.95364833623102E-2674

172.28846468403033E-2654

181.24975599133427E-2634

196.45926517626487E-2624

203.16826956895775E-260

211.47852579884703E-258

226.57943980486894E-257

232.79769223007052E-255

241.13889387865781E-253

254.44624170228036E-252

261.66734063835505E-250

276.01477696947373E-249

282.09013499689201E-247

297.00555592061776E-246

302.26746493297316E-244

317.09497091930352E-243

322.14844588150148E-241

336.30210791907139E-240

341.79239363462991E-238

354.94700643157885E-237

361.3260725573537E-235

373.45495660888891E-234

388.75558740621108E-233

392.15971156019862E-231

405.18870702337748E-230

411.2149167664493E-228

422.77405995005938E-227

436.1803475166436E-226

441.34422558487006E-224

452.85573257585713E-223

465.92874915205155E-222

471.20340993426742E-220

482.38927014032692E-219

494.64201055834918E-218

508.82910408198064E-217

511.64463703487867E-215

523.00146258865374E-214

535.36865383781807E-213

549.41502811928519E-212

551.61938483651697E-210

562.73271191162254E-209

574.52575446416059E-208

587.35825251673049E-207

591.17482607978979E-205

601.84251890180376E-204

612.83929142245154E-203

624.30015265432604E-202

636.40244950755179E-201

649.37358623215057E-200

651.3497964174296E-198

661.91221159135873E-197

672.66568003929694E-196

683.65746981862339E-195

694.94023459559014E-194

706.57051201213456E-193

718.6064453116697E-192

721.11047051313065E-190

731.41166662491137E-189

741.76839859634159E-188

752.18338280028321E-187

762.65740669771297E-186

773.18888803725574E-185

783.77351751075244E-184

794.40402929735944E-183

805.07013872858475E-182

815.75867608678799E-181

826.45393088263153E-180

837.13820307259769E-179

847.79253835425205E-178

858.39760603822981E-177

868.93466223834863E-176

879.38653021362197E-175

889.73852509663224E-174

899.9792526832912E-173

901.01012213271975E-171

911.01012213271981E-170

929.98044585480713E-170

939.74434928619933E-169

949.40226042827903E-168

958.96678731370675E-167

968.4530651238585E-166

977.87790811543057E-165

987.25892962064714E-164

996.6136914321448E-163

1005.95893598036281E-162

1015.30994295279823E-161

1024.68003795545682E-160

1034.08026610097085E-159

1043.51922951208756E-158

1053.00307585031455E-157

1062.53561593021856E-156

1072.1185426557152E-155

1081.75172091810535E-154

1091.43351840270632E-153

1101.1611499061922E-152

1119.31012086946846E-152

1127.38990844014101E-151

1135.80729088039367E-150

1144.51847983413112E-149

1153.48119402873039E-148

1162.65591096157462E-147

1172.00668828207852E-146

1181.50161504497918E-145

1191.1129617392198E-144

1208.17099410210585E-144

1215.94254116516754E-143

1224.28155220014963E-142

1233.05626246482209E-141

1242.16156627552349E-140

1251.5148256458868E-139

1261.05196225408811E-138

1277.23948826829104E-138

1284.93755723298257E-137

1293.33763558694655E-136

1302.23621584325406E-135

1311.48512044551996E-134

1329.77704293300596E-134

1336.38080696680427E-133

1344.128477343447E-132

1352.64834176253727E-131

1361.68442325337838E-130

1371.06229320505037E-129

1386.64318141998852E-129

1394.11972833383485E-128

1402.53363292530832E-127

1411.54533639415977E-126

1429.3481969195998E-126

1435.60891815176019E-125

1443.3380853167071E-124

1451.97062140075961E-123

1461.15402828606121E-122

1476.70435480473705E-122

1483.86406395164891E-121

1492.20951844752017E-120

1501.25353346589304E-119

1517.05631421198107E-119

1523.94132287234974E-118

1532.18447176192991E-117

1541.20145946906138E-116

1556.55764329565162E-116

1563.55205678514443E-115

1571.90951332908391E-114

1581.01880995975811E-113

1595.39520745985084E-113

1602.83585592108426E-112

1611.47957700230475E-111

1627.66274756131957E-111

1633.93949843950027E-110

1642.01058548406214E-109

1651.01869664525809E-108

1665.12416686018408E-108

1672.55901506670256E-107

1681.26884497057343E-106

1696.24662139359189E-106

1703.05349551651484E-105

1711.48210601093987E-104

1727.14340629691407E-104

1733.41892509470781E-103

1741.62497186972616E-102

1757.66986722510707E-102

1763.59525026176913E-101

1771.67372102581788E-100

1787.73860901263028E-100

1793.55370760244789E-99

1801.62088552311661E-98

1817.34323828152231E-98

1823.30445722668522E-97

1831.4770743231849E-96

1846.55853109805506E-96

1852.89284398703385E-95

1861.26756335991006E-94

1875.51762874313763E-94

1882.38608094051629E-93

1891.02513107074042E-92

1904.37569104405492E-92

1911.85565955271449E-91

1927.81889884451013E-91

1933.27340428309043E-90

1941.36166868889373E-89

1955.62823058076105E-89

1962.31159470281244E-88

1979.43412254345842E-88

1983.82606080929125E-87

1991.54196018545315E-86

2006.1755505427395E-86

2012.45793056427457E-85

2029.72221049928351E-85

2033.82183447213234E-84

2041.49313827171049E-83

2055.79774665503228E-83

2062.23747989842252E-82

2078.58241081810427E-82

2083.27204412440206E-81

2091.2399325102998E-80

2104.67041245546233E-80

2111.74863783877509E-79

2126.5079021452522E-79

2132.40761825843147E-78

2148.85418490367036E-78

2153.23692527176061E-77

2161.17638256404256E-76

2174.2501563604116E-76

2181.5265469863314E-75

2195.45095773201427E-75

2201.93508999486517E-74

2216.8297293936414E-74

2222.39655819713827E-73

2238.36108644562095E-73

2242.90025186082493E-72

2251.00026464177778E-71

2263.43011105034437E-71

2271.16956209381779E-70

2283.96522587070704E-70

2291.33674863414221E-69

2304.48101389966827E-69

2311.49367129988934E-68

2324.95100530006454E-68

2331.63191934354049E-67

2345.34906895938316E-67

2351.74356886080309E-66

2365.65182279031544E-66

2371.82193781088639E-65

2385.8409182760773E-65

2391.86225093153583E-64

2405.90488732874521E-64

2411.86212214516436E-63

2425.84029218256127E-63

2431.82178661497168E-62

2445.65201830956412E-62

2451.74405136409397E-61

2465.35267796703668E-61

2471.6339753794111E-60

2484.96122363184068E-60

2491.49832938600181E-59

2504.50098147554919E-59

2511.34491478353071E-58

2523.99738560660495E-58

2531.18183574456153E-57

2543.47571378420241E-57

2551.01681666000594E-56

2562.95909535822023E-56

2578.56640835220226E-56

2582.46699279290152E-55

2597.06760097425881E-55

2602.01426627766365E-54

2615.71094653437235E-54

2621.61083568278671E-53

2634.52013967261033E-53

2641.26187232527042E-52

2653.50467181660011E-52

2669.68396159850033E-52

2672.6621827016102E-51

2687.28126835925478E-51

2691.98137116690501E-50

2705.3643789741021E-50

2711.44501721442603E-49

2723.87285863719329E-49

2731.03276230325154E-48

2742.74021238855427E-48

2757.2341607057833E-48

2761.90027772162786E-47

2774.96679086808148E-47

2781.29172294878522E-46

2793.34273824022564E-46

2808.60755096858103E-46

2812.2054934866115E-45

2825.62322630097048E-45

2831.42667013572325E-44

2843.60183974406187E-44

2859.04883247981859E-44

2862.26220811995465E-43

2875.62793239598475E-43

2881.39330409664483E-42

2893.43263846647446E-42

2908.41588258504601E-42

2912.05335966851638E-41

2924.9857260444456E-41

2931.20474199299225E-40

2942.89711764981477E-40

2956.93344088396349E-40

2961.65137696729535E-39

2973.914375033589E-39

2989.23424714299683E-39

2992.16804063357318E-38

3005.06598828044931E-38

3011.17813680940682E-37

3022.7268795687926E-37

3036.28172257101397E-37

3041.44025020789367E-36

3053.28660375309506E-36

3067.46467192287929E-36

3071.68745352263134E-35

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3098.50279978879314E-35

3101.89530150130841E-34

3114.20500976174533E-34

3129.28606322385428E-34

3132.04115383323059E-33

3144.46583657143127E-33

3159.72559964445029E-33

3162.10823916343305E-32

3174.54900816336973E-32

3189.7703540112627E-32

3192.08883430585616E-31

3204.44530050715015E-31

3219.41683596530249E-31

3221.98572410572683E-30

3234.16817629623132E-30

3248.70943010045888E-30

3251.81156146089545E-29

3263.75093247271297E-29

3277.73127977556128E-29

3281.58632661248559E-28

3293.24015648507696E-28

3306.58831818632316E-28

3311.33358706490529E-27

3322.68725827235434E-27

3335.39065623403212E-27

3341.07651727787408E-26

3352.14018061810191E-26

3364.23577413999334E-26

3378.34585765268717E-26

3381.6370720780271E-25

3393.19687821726835E-25

3406.21510735768935E-25

3411.20292400471407E-24

3422.31791496814787E-24

3434.4466123878755E-24

3448.49251261289013E-24

3451.61480819537853E-23

3463.05693458951717E-23

3475.76148478831189E-23

3481.08110619734703E-22

3492.01971702197783E-22

3503.75667366087877E-22

3516.9568030757014E-22

3521.28266056708245E-21

3532.3545723724345E-21

3544.30341334735921E-21

3557.83100006308185E-21

3561.41881883165388E-20

3572.55943789239523E-20

3584.5969792313132E-20

3598.22078180084424E-20

3601.46375587065032E-19

3612.59502425821664E-19

3624.58071961602329E-19

3638.05096174937426E-19

3641.4089183061405E-18

3652.4549918978229E-18

3664.25934386644137E-18

3677.35810357308944E-18

3681.26567379395805E-17

3692.16776649805287E-17

3703.696920703436E-17

3716.27778987375923E-17

3721.06148651360069E-16

3731.7871676422017E-16

3742.99613398839696E-16

3755.001546337964E-16

3768.313740588371E-16

3770

3780

3790

3800

3810

3820

3830

3840

3850

3860

3870

3880

3890

3900

3910

3920

3930

3940

3950

3960

3970

3980

3990

4000

4010.0000000001

4020.0000000001

4030.0000000002

4040.0000000002

4050.0000000003

4060.0000000005

4070.0000000007

4080.000000001

4090.0000000015

4100.0000000022

4110.0000000031

4120.0000000044

4130.0000000063

4140.000000009

4150.0000000126

4160.0000000178

4170.0000000249

4180.0000000347

4190.0000000483

4200.0000000668

4210.000000092

4220.0000001262

4230.0000001724

4240.0000002347

4250.000000318

4260.0000004293

4270.000000577

4280.0000007725

4290.0000010301

4300.0000013678

4310.0000018089

4320.0000023826

4330.0000031255

4340.0000040833

4350.0000053129

4360.0000068849

4370.0000088858

4380.0000114216

4390.0000146218

4400.0000186428

4410.0000236734

4420.0000299398

4430.000037712

4440.0000473099

4450.0000591108

4460.0000735572

4470.0000911648

4480.0001125316

4490.0001383462

4500.0001693972

4510.000206582

4520.0002509149

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&A

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Sheet1

&A

Page &P

Number

Probability

Dice

Sheet2

&A

Page &P

Successes

Probabilty


Sheet3

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Calc

Exp

No of Heads

No of occurences


Sheet4


p =0.4Sqrt(Npq) =4.381780462.5066282746

q =0.6Mean =Np =3210.9834947944

N =800.09104570250.0910457025

nBinomialGaussian

01.78689910246017E-180

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&A

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Sheet4

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Page &P

Binomial

Gaussian

No. of Heads

Probability

Both

Sheet5

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Page &P

No. of Heads

Probability

Gaussian

Sheet6

&A

Page &P

No. of Heads

Probability

Binomial

Sheet7

&A

Page &P

No. of Heads

Probability

Gaussian

Sheet8

&A

Page &P

Binomial

Gaussian

No. of Heads

Probability

Both

Sheet9

&A

Page &P

Sheet10

&A

Page &P

Sheet11

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Page &P

Sheet12

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Page &P

Sheet13

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Page &P

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Sheet16

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Multiple Outcomes


Stirlings Approximation


Number ExpectedToss 6 coins N times. Probability of n heads:

Number of times n heads is expected is:n = N P(n)


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Thermodynamics and Statistical Mechanics

Documents

dicethermo stat mech

class 16chart10

desired outcome

total number of outcomes

pair of dicelist

pair of dicefor

particular outcome

total combinationshow