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# What is the probability of this happening again?

Topic closed. 18 replies. Last post 8 years ago by truecritic.

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Stone Mountain*Georgia
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 Posted: September 30, 2009, 12:06 pm - IP Logged

Odds never change ......but the probabilities do.

-----------------------------------------------------------

Say ..... can you toss a coin 33 times in a row ......and never get 2 heads in a row ?

Well, Ga Eve Pick 3 game can.

Heads and Tails ..... Odd/Even  High/LOw

Mostly to all High // Mostly to all Lows            500 of each out of the P3 1000 right ?

Ga Eve has not had back to back Mostly to all  Low ........ since Aug 27th !   Today is Sept 30Th. !

So..... for the last 30 + draws ....... there has been NO back to back Heads (All to Mostly Low)

50/50 odds each day every day .......

......and yet......no 2 draws in a row of Mostly to All low digits.

Whats the probability of that happening again next month?

The only real failure .....is the failure to try.

Luck is a very rare thing....... Odds not so much.

Odds never change .....but probability does.

Win d

NY
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 Posted: October 1, 2009, 2:22 am - IP Logged

All high/ all low isn't a 500/500 proposition. High and low each have 5 digits, so there are 125 (5*5*5) numbers that qualify. That means that you should expect all high (as well as all low, all odd, or all even) in 1 of every 8 drawings. Since the odds of having it happening again in the next drawing would also be 1 in 8, we would expect it to happen twice in a row once every 64 drawings. Compare that to once in every 4 for flipping a coin.

The probability that it won't happen for any number of  drawings is the probability of a different outcome, raised to the nth power, where n is the number of drawings. The probability of a different outcome is 63/64, or 0.984375, so the probability that it won't happen in 31 drawings is 0.984375 to the 30th power*. That's 0.623, or 62.3%. Conversely there's a 37.7% chance that it will happen in any given month.

I'm not sure what "mostly to all high" means. If it means 2 high and whatever the 3rd digit is, then there are 5*5*10 or 250 numbers that qualify. For that we'd expect repeats once every 16 drawings. The probability of not happening in a month would be 15/16  or 0.9375 ^ 30 or about 14.4%. That's almost two months per year.

* I used the 30th power instead of 31st, because there can't be a repeat on the first drawings, and if there's a repeat after the 31st drawing it will be in the next month. The difference for raising to the 31st is very small anyway.

Stone Mountain*Georgia
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 Posted: October 1, 2009, 4:27 pm - IP Logged

Hello Ky Floyd ....good to know your around.  This was a confusing term to someone not used to PICK 3 language. Sorry about that.  I used a term which is pretty well known within the Pick 3 Hobby Player community.... " Mostly to ALL HIGH" ...Mostly to ALL LOW.....  Mostly to all Even .....Mostly to all ODD.

What this pretty commonly used  phrase indicates to mostly to ALL of us is.... LOL   The 10 pure numbers in each of those groups...... as well as the numbers which contain 2 of the named group digits.

For example :

The 10 single number ALL HIGH ....... 569, 578, 579, 678, 589, 679, 689, 789, 567, 568

Along with the remaining 50 Mostly single like numbers which contain at least 2 of the high digits

389, 479, 056, 489, 057, 156, 058, 067, 157, 256, 059, 068, 158, 167, 257, 356, 069, 078, 159, 168, 258, 267, 357, 456, 079, 169, 178, 259, 268, 358, 367, 457, 089, 179, 269, 278, 359, 368, 458, 467, 189, 279, 369, 378, 459, 468, 289, 379, 469, 478

All toll...... 60 singles Box numbers ....or ..... 360 straights

Double number versions of ALL HIGH .....20 of them

668, 677, 588, 669, 688, 778, 599, 779, 788, 699,

799, 889, 556, 899, 557, 566, 558, 559, 577, 667

Mostly Highs  ...... 25 doubles

055, 299, 488, 155, 399, 066, 255, 499, 166, 355,

077,

266, 455, 177, 366, 088, 277, 466, 188, 377, 099, 288, 477, 199, 388

Plus ......the 5 ALL HIGH Trips ....555 666 777 888 999

All toll..... 45 doubles or 135 straights .......plus 5 trips

360 singles 135 doubles .....5 trips =    500 Straights

===========================================

All of these Box types above translate to these numbers below...... 495 of them plus the 5 trips versions for a total of500 numbers .   Of course the Mostly to ALL LOW numbers would be the other HALF .

555 666 777 888 999 +

055, 056, 057, 058, 059, 065, 066, 067, 068, 069, 075, 076, 077, 078, 079, 085, 086, 087, 088, 089, 095, 096, 097, 098, 099, 155, 156, 157, 158, 159, 165, 166, 167, 168, 169, 175, 176, 177, 178, 179, 185, 186, 187, 188, 189, 195, 196, 197, 198, 199, 255, 256, 257, 258, 259, 265, 266, 267, 268, 269, 275, 276, 277, 278, 279, 285, 286, 287, 288, 289, 295, 296, 297, 298, 299, 355, 356, 357, 358, 359, 365, 366, 367, 368, 369, 375, 376, 377, 378, 379, 385, 386, 387, 388, 389, 395, 396, 397, 398, 399, 455, 456, 457, 458, 459, 465, 466, 467, 468, 469, 475, 476, 477, 478, 479, 485, 486, 487, 488, 489, 495, 496, 497, 498, 499, 505, 506, 507, 508, 509, 515, 516, 517, 518, 519, 525, 526, 527, 528, 529, 535, 536, 537, 538, 539, 545, 546, 547, 548, 549, 550, 551, 552, 553, 554, 556, 557, 558, 559, 560, 561, 562, 563, 564, 565, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576, 577, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589, 590, 591, 592, 593, 594, 595, 596, 597, 598, 599, 605, 606, 607, 608, 609, 615, 616, 617, 618, 619, 625, 626, 627, 628, 629, 635, 636, 637, 638, 639, 645, 646, 647, 648, 649, 650, 651, 652, 653, 654, 655, 656, 657, 658, 659, 660, 661, 662, 663, 664, 665, 667, 668, 669, 670, 671, 672, 673, 674, 675, 676, 677, 678, 679, 680, 681, 682, 683, 684, 685, 686, 687, 688, 689, 690, 691, 692, 693, 694, 695, 696, 697, 698, 699, 705, 706, 707, 708, 709, 715, 716, 717, 718, 719, 725, 726, 727, 728, 729, 735, 736, 737, 738, 739, 745, 746, 747, 748, 749, 750, 751, 752, 753, 754, 755, 756, 757, 758, 759, 760, 761, 762, 763, 764, 765, 766, 767, 768, 769, 770, 771, 772, 773, 774, 775, 776, 778, 779, 780, 781, 782, 783, 784, 785, 786, 787, 788, 789, 790, 791, 792, 793, 794, 795, 796, 797, 798, 799, 805, 806, 807, 808, 809, 815, 816, 817, 818, 819, 825, 826, 827, 828, 829, 835, 836, 837, 838, 839, 845, 846, 847, 848, 849, 850, 851, 852, 853, 854, 855, 856, 857, 858, 859, 860, 861, 862, 863, 864, 865, 866, 867,868, 869, 870, 871, 872, 873, 874, 875, 876, 877, 878, 879, 880, 881, 882, 883, 884, 885, 886, 887, 889, 890, 891, 892, 893, 894, 895, 896, 897, 898, 899, 905, 906, 907, 908, 909, 915, 916, 917, 918, 919, 925, 926, 927, 928, 929, 935, 936, 937, 938, 939, 945, 946, 947, 948, 949, 950, 951, 952, 953, 954, 955, 956, 957, 958, 959, 960, 961, 962, 963, 964, 965, 966, 967, 968, 969, 970, 971, 972, 973, 974, 975, 976, 977, 978, 979, 980, 981, 982, 983, 984, 985, 986, 987, 988, 989, 990, 991, 992, 993, 994, 995, 996, 997, 998

The reason I highlighted the 868 is because Ga Eve last night drew another ...." Mostly to ALL HIGH" number from that half of the 1000 chart.

Now 34 days or Coin flips...... and No back to back Tails from the  Mostly to ALL LOW half of the chart..

So ..... the beat goes on for one more night without hitting 2 Heads in a row.

******  One thing we do know for sure .....beyond All doubt and with 100% surity....  It will have to go on to at least draw 36 now ...before this String ends. That much is sure.

The only real failure .....is the failure to try.

Luck is a very rare thing....... Odds not so much.

Odds never change .....but probability does.

Win d

Stone Mountain*Georgia
United States
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 Posted: October 1, 2009, 4:51 pm - IP Logged

The 500 "Mostly to ALL LOW Group"  or  The Tails side of the Coin .  50/50

Pure ALL LOWS  ....10 singles +20 doubles and 5 trips 000 111 222 333 444

244, 334, 001, 344, 002, 011, 003, 012, 004, 013, 022, 112, 014, 023, 113, 122, 024, 033, 114, 123, 034, 124, 133, 223, 044, 134, 224, 233, 144, 234

The Mostly Low Group... 375 of them

005, 006, 007, 008, 009, 015, 016, 017, 018, 019, 025, 026, 027, 028, 029, 035, 036, 037, 038, 039, 045, 046, 047, 048, 049, 050, 051, 052, 053, 054, 060, 061, 062, 063, 064, 070, 071, 072, 073, 074, 080, 081, 082, 083, 084, 090, 091, 092, 093, 094, 105, 106, 107, 108, 109, 115, 116, 117, 118, 119, 125, 126, 127, 128, 129, 135, 136, 137, 138, 139, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 160, 161, 162, 163, 164, 170, 171, 172, 173, 174, 180, 181, 182, 183, 184, 190, 191, 192, 193, 194, 205, 206, 207, 208, 209, 215, 216, 217, 218, 219, 225, 226, 227, 228, 229, 235, 236, 237, 238, 239, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 260, 261, 262, 263, 264, 270, 271, 272, 273, 274, 280, 281, 282, 283, 284, 290, 291, 292, 293, 294, 305, 306, 307, 308, 309, 315, 316, 317, 318, 319, 325, 326, 327, 328, 329, 335, 336, 337, 338, 339, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 360, 361, 362, 363, 364, 370, 371, 372, 373, 374, 380, 381, 382, 383, 384, 390, 391, 392, 393, 394, 405, 406, 407, 408, 409, 415, 416, 417, 418, 419, 425, 426, 427, 428, 429, 435, 436, 437, 438, 439, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 460, 461, 462, 463, 464, 470, 471, 472, 473, 474, 480, 481, 482, 483, 484, 490, 491, 492, 493, 494, 500, 501, 502, 503, 504, 510, 511, 512, 513, 514, 520, 521, 522, 523, 524, 530, 531, 532, 533, 534, 540, 541, 542, 543, 544, 600, 601, 602, 603, 604, 610, 611, 612, 613, 614, 620, 621, 622, 623, 624, 630, 631, 632, 633, 634, 640, 641, 642, 643, 644, 700, 701, 702, 703, 704, 710, 711, 712, 713, 714, 720, 721, 722, 723, 724, 730, 731, 732, 733, 734, 740, 741, 742, 743, 744, 800, 801, 802, 803, 804, 810, 811, 812, 813, 814, 820, 821, 822, 823, 824, 830, 831, 832, 833, 834, 840, 841, 842, 843, 844, 900, 901, 902, 903, 904, 910, 911, 912, 913, 914, 920, 921, 922, 923, 924, 930, 931, 932, 933, 934, 940, 941, 942, 943, 944

Now, the question is going to have to be ...... What is the probability of tossing a coin 36 times without getting 2 Tails in a row ?    At least for now....thats the question. LOL

The only real failure .....is the failure to try.

Luck is a very rare thing....... Odds not so much.

Odds never change .....but probability does.

Win d

Stone Mountain*Georgia
United States
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 Posted: October 1, 2009, 8:05 pm - IP Logged

059 hit tonight ....... from the Mostly to ALL High group ...... again.  Mostly Odd too.

Now the question will have to be.... What is the probability of tossing a coin 38 times without getting 2 Tails in a row?

I guess a good way to frame this is to take it as ......( 2 tosses equal = 1 event)  So far we have had 18 2 toss events .....and 18 negative outcomes. If tomorrow is also another Mostly to ALL High..... it will be 19 negative outcomes in a row.  wow...

Someone betting the other way would have lost 18 two game events in a row so far!  So to speak... and if tomorrow is another Mostly to all High number ....it wil. be 19 losses in a row.

=================================================================

.

All of these Box types above translate to these numbers below...... 495 of them plus the 5 trips versions for a total of 500 numbers .   Of course the Mostly to ALL LOW numbers would be the other HALF .

555 666 777 888 999 +

055, 056, 057, 058,059, 065, 066, 067, 068, 069, 075, 076, 077, 078, 079, 085, 086, 087, 088, 089, 095, 096, 097, 098, 099, 155, 156, 157, 158, 159, 165, 166, 167, 168, 169, 175, 176, 177, 178, 179, 185, 186, 187, 188, 189, 195, 196, 197, 198, 199, 255, 256, 257, 258, 259, 265, 266, 267, 268, 269, 275, 276, 277, 278, 279, 285, 286, 287, 288, 289, 295, 296, 297, 298, 299, 355, 356, 357, 358, 359, 365, 366, 367, 368, 369, 375, 376, 377, 378, 379, 385, 386, 387, 388, 389, 395, 396, 397, 398, 399, 455, 456, 457, 458, 459, 465, 466, 467, 468, 469, 475, 476, 477, 478, 479, 485, 486, 487, 488, 489, 495, 496, 497, 498, 499, 505, 506, 507, 508, 509, 515, 516, 517, 518, 519, 525, 526, 527, 528, 529, 535, 536, 537, 538, 539, 545, 546, 547, 548, 549, 550, 551, 552, 553, 554, 556, 557, 558, 559, 560, 561, 562, 563, 564, 565, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576, 577, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589, 590, 591, 592, 593, 594, 595, 596, 597, 598, 599, 605, 606, 607, 608, 609, 615, 616, 617, 618, 619, 625, 626, 627, 628, 629, 635, 636, 637, 638, 639, 645, 646, 647, 648, 649, 650, 651, 652, 653, 654, 655, 656, 657, 658, 659, 660, 661, 662, 663, 664, 665, 667, 668, 669, 670, 671, 672, 673, 674, 675, 676, 677, 678, 679, 680, 681, 682, 683, 684, 685, 686, 687, 688, 689, 690, 691, 692, 693, 694, 695, 696, 697, 698, 699, 705, 706, 707, 708, 709, 715, 716, 717, 718, 719, 725, 726, 727, 728, 729, 735, 736, 737, 738, 739, 745, 746, 747, 748, 749, 750, 751, 752, 753, 754, 755, 756, 757, 758, 759, 760, 761, 762, 763, 764, 765, 766, 767, 768, 769, 770, 771, 772, 773, 774, 775, 776, 778, 779, 780, 781, 782, 783, 784, 785, 786, 787, 788, 789, 790, 791, 792, 793, 794, 795, 796, 797, 798, 799, 805, 806, 807, 808, 809, 815, 816, 817, 818, 819, 825, 826, 827, 828, 829, 835, 836, 837, 838, 839, 845, 846, 847, 848, 849, 850, 851, 852, 853, 854, 855, 856, 857, 858, 859, 860, 861, 862, 863, 864, 865, 866, 867,868, 869, 870, 871, 872, 873, 874, 875, 876, 877, 878, 879, 880, 881, 882, 883, 884, 885, 886, 887, 889, 890, 891, 892, 893, 894, 895, 896, 897, 898, 899, 905, 906, 907, 908, 909, 915, 916, 917, 918, 919, 925, 926, 927, 928, 929, 935, 936, 937, 938, 939, 945, 946, 947, 948, 949, 950, 951, 952, 953, 954, 955, 956, 957, 958, 959, 960, 961, 962, 963, 964, 965, 966, 967, 968, 969, 970, 971, 972, 973, 974, 975, 976, 977, 978, 979, 980, 981, 982, 983, 984, 985, 986, 987, 988, 989, 990, 991, 992, 993, 994, 995, 996, 997, 998

The only real failure .....is the failure to try.

Luck is a very rare thing....... Odds not so much.

Odds never change .....but probability does.

Win d

NY
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 Posted: October 2, 2009, 1:16 am - IP Logged

What is the probability of tossing a coin 38 times without getting 2 Tails in a row?

I guess a good way to frame this is to take it as ......( 2 tosses equal = 1 event)  So far we have had 18 2 toss events

Nope. Tossing a coin 38 times gives you 37 "2 toss events." Each  event starts with the first of 2 consecutive tosses, and ends with the following toss. Toss 1 starts the first event and toss 2 finishes it, but #2 is also the start of the 2nd event:
1 & 2
2 & 3
3 & 4
.
.
.
37 & 38

Toss 38 won't become a 2 toss event until toss 39 occurs.

The probability of a result other than two tails in any event is .75. Just as the probability ot two tails is .5 x .5, the probability of two events both being something other than two tails is .75 x .75.  For n consecutive events it's .756^n.  For the 37 events *ending* with toss 38 it would be .75^37, which is .000024 or 1 in 41,950.

The Quantum Master
West Concord, MN
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 Posted: October 2, 2009, 10:14 am - IP Logged

It's not quite that simple an equation. The progression of No Back-to-Back and Back-to-Back Bit Flips or Coin Tosses is as follows:

 Bit Flips or Possible Outcomes Odds Probability of Probability of Coin Tosses No Back-to-Back Back-to-Back NBB : BB No Back-to-Back Back-to-Back 2 3 1 1 : 0.33 75.0000% 25.0000% 3 5 3 1 : 0.60 62.5000% 37.5000% 4 8 8 1 : 1.00 50.0000% 50.0000% 5 13 19 1 : 1.46 40.6250% 59.3750% 6 21 43 1 : 2.05 32.8125% 67.1875% 7 34 94 1 : 2.76 26.5625% 73.4375% 8 55 201 1 : 3.65 21.4844% 78.5156% 9 89 423 1 : 4.75 17.3828% 82.6172% 10 144 880 1 : 6.11 14.0625% 85.9375% 11 233 1815 1 : 7.79 11.3770% 88.6230% 12 377 3719 1 : 9.86 9.2041% 90.7959% 13 610 7582 1 : 12.43 7.4463% 92.5537% 14 987 15397 1 : 15.60 6.0242% 93.9758% 15 1597 31171 1 : 19.52 4.8737% 95.1263% 16 2584 62952 1 : 24.36 3.9429% 96.0571% 17 4181 126891 1 : 30.35 3.1898% 96.8102% 18 6765 255379 1 : 37.75 2.5806% 97.4194% 19 10946 513342 1 : 46.90 2.0878% 97.9122% 20 17711 1030865 1 : 58.20 1.6891% 98.3109% 21 28657 2068495 1 : 72.18 1.3665% 98.6335% 22 46368 4147936 1 : 89.46 1.1055% 98.8945% 23 75025 8313583 1 : 110.81 0.8944% 99.1056% 24 121393 16655823 1 : 137.21 0.7236% 99.2764% 25 196418 33358014 1 : 169.83 0.5854% 99.4146% 26 317811 66791053 1 : 210.16 0.4736% 99.5264% 27 514229 133703499 1 : 260.01 0.3831% 99.6169% 28 832040 267603416 1 : 321.62 0.3100% 99.6900% 29 1346269 535524643 1 : 397.78 0.2508% 99.7492% 30 2178309 1071563515 1 : 491.92 0.2029% 99.7971% 31 3524578 2143959070 1 : 608.29 0.1641% 99.8359% 32 5702887 4289264409 1 : 752.12 0.1328% 99.8672% 33 9227465 8580707127 1 : 929.91 0.1074% 99.8926% 34 14930352 17164938832 1 : 1149.67 0.0869% 99.9131% 35 24157817 34335580551 1 : 1421.30 0.0703% 99.9297% 36 39088169 68680388567 1 : 1757.06 0.0569% 99.9431% 37 63245986 137375707486 1 : 2172.09 0.0460% 99.9540% 38 102334155 274775572789 1 : 2685.08 0.0372% 99.9628% 39 165580141 549590233747 1 : 3319.18 0.0301% 99.9699% 40 267914296 1099243713480 1 : 4102.97 0.0244% 99.9756% 41 433494437 2198589761115 1 : 5071.78 0.0197% 99.9803% 42 701408733 4397345102371 1 : 6269.30 0.0159% 99.9841% 43 1134903170 8794958119038 1 : 7749.52 0.0129% 99.9871% 44 1836311903 17590349732513 1 : 9579.17 0.0104% 99.9896% 45 2971215073 35181400873759 1 : 11840.75 0.0084% 99.9916% 46 4807526976 70363936650688 1 : 14636.20 0.0068% 99.9932% 47 7778742049 140729709613279 1 : 18091.58 0.0055% 99.9945% 48 12586269025 281462390441631 1 : 22362.65 0.0045% 99.9955% 49 20365011074 562929588410238 1 : 27642.00 0.0036% 99.9964% 50 32951280099 1125866955562520 1 : 34167.62 0.0029% 99.9971%

Presented 'AS IS' and for Entertainment Purposes Only.
Any gain or loss is your responsibility.

Order is a Subset of Chaos
Knowledge is Beyond Belief
Wisdom is Not Censored
Douglas Paul Smallish
Jehocifer

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 Posted: October 2, 2009, 2:56 pm - IP Logged

It is that simple. What you've done is list all possible outcomes and the ratio between two types of outcomes, and assumed that the ratio  can be used to determine the probabilty of one unlikely set of possible outcomes. That works for the possible outcomes of a single event, but not multiple iterations, as in this case. In essence you're using each new flip to change the probability of previous flips.

Let's  start from flip # 2 and then look at what happens with flip # 3. As you correctly note, after flip # 2 there are 4 possible outcomes, and there's a 75% chance that we've had one of the 3 results that didn't already give us back to back tails. Since we're looking at the probability of not having any back to back tails, we're only concerned with outcomes where that's the result we've gotten, and that 4th possible result is no longer relevant. If it happened it happened, and it's now a certainty rather than a probability. Therefore we've conducted the 2nd flip, and it was either heads or tails, with a 50% chance of each. We now make flip # 3, and there is a 50% chance it will be heads and a 50% chance it will be tails. That means that when we see the result of flip #3 there is a 75% chance that we will have ended our streak with no back to back tails and a 25% chance that it will remain unbroken.

That means that after flip #3 the chance that we still don't have back to back tails is 75% of what it was after flip #2. 75% of 75% is 56.25%, not the 62.5% you've shown.

As we move down the chart we'll have the same pattern. Each succeeding flip may increase the total number of possible outcomes that could have happened by a factor of two, but it only adds 4 new possibilities to the results we've actually had. Each time we reduce the possibility of no back to back tails to 75% of the previous possibility. Your figures consistently show a figure that is close to 80%, but varies a bit.

BTW, in # 50 you're missing 5 possible outcomes in the no back to back column.

The Quantum Master
West Concord, MN
United States
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December 7, 2001
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 Posted: October 2, 2009, 4:24 pm - IP Logged

It is that simple. What you've done is list all possible outcomes and the ratio between two types of outcomes, and assumed that the ratio  can be used to determine the probabilty of one unlikely set of possible outcomes. That works for the possible outcomes of a single event, but not multiple iterations, as in this case. In essence you're using each new flip to change the probability of previous flips.

Let's  start from flip # 2 and then look at what happens with flip # 3. As you correctly note, after flip # 2 there are 4 possible outcomes, and there's a 75% chance that we've had one of the 3 results that didn't already give us back to back tails. Since we're looking at the probability of not having any back to back tails, we're only concerned with outcomes where that's the result we've gotten, and that 4th possible result is no longer relevant. If it happened it happened, and it's now a certainty rather than a probability. Therefore we've conducted the 2nd flip, and it was either heads or tails, with a 50% chance of each. We now make flip # 3, and there is a 50% chance it will be heads and a 50% chance it will be tails. That means that when we see the result of flip #3 there is a 75% chance that we will have ended our streak with no back to back tails and a 25% chance that it will remain unbroken.

That means that after flip #3 the chance that we still don't have back to back tails is 75% of what it was after flip #2. 75% of 75% is 56.25%, not the 62.5% you've shown.

As we move down the chart we'll have the same pattern. Each succeeding flip may increase the total number of possible outcomes that could have happened by a factor of two, but it only adds 4 new possibilities to the results we've actually had. Each time we reduce the possibility of no back to back tails to 75% of the previous possibility. Your figures consistently show a figure that is close to 80%, but varies a bit.

BTW, in # 50 you're missing 5 possible outcomes in the no back to back column.

You're right about the 50 Back-to-Back flips; it's an Excel computation issue. It actually should be 1125866955562525. It’s caused by the fact that Excel tunicates the number after a certain number of digits; in this case 15 digits and the 5 was dropped.

Buuut... you are wrong about the calculation of probability; in this case, the probability of No Back-to-Back. Where you went wrong is not considering ALL possibilities. That's what the table shows. So, let's go through the progression. We'll use 1's and 0's to represent the flip; where 1 is outcome occurred and 0 is the outcome did not. Since this is a succession of flips, we'll start at 2 flips and look at ALL the possible outcomes. Now, by definition, a No Back-to-Back flip is when there are no consecutive 1’s; i.e. (1 1). The following is the total possible outcomes for 2 flips.

 Flip # Outcome # 1 2 No Back-To-Back 1 0 0 TRUE 2 0 1 TRUE 3 1 0 TRUE 4 1 1 FALSE

From this we can see there are 3 possible outcomes where there are No Back-to-Back flips. Given that the total number of outcomes is 4, the probability of having a No Back-to-Back flip is 3 / 4, or 75%. Continuing this progression, we can go to 3 flips as follows:

 Flip # Outcome # 1 2 3 No Back-To-Back 1 0 0 0 TRUE 2 0 0 1 TRUE 3 0 1 0 TRUE 4 0 1 1 FALSE 5 1 0 0 TRUE 6 1 0 1 TRUE 7 1 1 0 FALSE 8 1 1 1 FALSE

The table shows there are 5 possible outcomes where there is No Back-to-Back flip. This becomes 5 / 8, or 62.5% probability of having a No Back-to-Back flip. The progression continues...

 Flip # Outcome # 1 2 3 4 No Back-To-Back 1 0 0 0 0 TRUE 2 0 0 0 1 TRUE 3 0 0 1 0 TRUE 4 0 0 1 1 FALSE 5 0 1 0 0 TRUE 6 0 1 0 1 TRUE 7 0 1 1 0 FALSE 8 0 1 1 1 FALSE 9 1 0 0 0 TRUE 10 1 0 0 1 TRUE 11 1 0 1 0 TRUE 12 1 0 1 1 FALSE 13 1 1 0 0 FALSE 14 1 1 0 1 FALSE 15 1 1 1 0 FALSE 16 1 1 1 1 FALSE

In the case of 4 flips, there are 8 out of 16 possible outcomes for a probability of 50%. Moving along...

 Flip # Outcome # 1 2 3 4 5 No Back-To-Back 1 0 0 0 0 0 TRUE 2 0 0 0 0 1 TRUE 3 0 0 0 1 0 TRUE 4 0 0 0 1 1 FALSE 5 0 0 1 0 0 TRUE 6 0 0 1 0 1 TRUE 7 0 0 1 1 0 FALSE 8 0 0 1 1 1 FALSE 9 0 1 0 0 0 TRUE 10 0 1 0 0 1 TRUE 11 0 1 0 1 0 TRUE 12 0 1 0 1 1 FALSE 13 0 1 1 0 0 FALSE 14 0 1 1 0 1 FALSE 15 0 1 1 1 0 FALSE 16 0 1 1 1 1 FALSE 17 1 0 0 0 0 TRUE 18 1 0 0 0 1 TRUE 19 1 0 0 1 0 TRUE 20 1 0 0 1 1 FALSE 21 1 0 1 0 0 TRUE 22 1 0 1 0 1 TRUE 23 1 0 1 1 0 FALSE 24 1 0 1 1 1 FALSE 25 1 1 0 0 0 FALSE 26 1 1 0 0 1 FALSE 27 1 1 0 1 0 FALSE 28 1 1 0 1 1 FALSE 29 1 1 1 0 0 FALSE 30 1 1 1 0 1 FALSE 31 1 1 1 1 0 FALSE 32 1 1 1 1 1 FALSE

5 flips, there are 13 out of 32 possible, this works to 40.625% probability of having No Back-to-Back flips. And, one more to establish the proper sequence.

 Flip # Outcome # 1 2 3 4 5 6 No Back-To-Back 1 0 0 0 0 0 0 TRUE 2 0 0 0 0 0 1 TRUE 3 0 0 0 0 1 0 TRUE 4 0 0 0 0 1 1 FALSE 5 0 0 0 1 0 0 TRUE 6 0 0 0 1 0 1 TRUE 7 0 0 0 1 1 0 FALSE 8 0 0 0 1 1 1 FALSE 9 0 0 1 0 0 0 TRUE 10 0 0 1 0 0 1 TRUE 11 0 0 1 0 1 0 TRUE 12 0 0 1 0 1 1 FALSE 13 0 0 1 1 0 0 FALSE 14 0 0 1 1 0 1 FALSE 15 0 0 1 1 1 0 FALSE 16 0 0 1 1 1 1 FALSE 17 0 1 0 0 0 0 TRUE 18 0 1 0 0 0 1 TRUE 19 0 1 0 0 1 0 TRUE 20 0 1 0 0 1 1 FALSE 21 0 1 0 1 0 0 TRUE 22 0 1 0 1 0 1 TRUE 23 0 1 0 1 1 0 FALSE 24 0 1 0 1 1 1 FALSE 25 0 1 1 0 0 0 FALSE 26 0 1 1 0 0 1 FALSE 27 0 1 1 0 1 0 FALSE 28 0 1 1 0 1 1 FALSE 29 0 1 1 1 0 0 FALSE 30 0 1 1 1 0 1 FALSE 31 0 1 1 1 1 0 FALSE 32 0 1 1 1 1 1 FALSE 33 1 0 0 0 0 0 TRUE 34 1 0 0 0 0 1 TRUE 35 1 0 0 0 1 0 TRUE 36 1 0 0 0 1 1 FALSE 37 1 0 0 1 0 0 TRUE 38 1 0 0 1 0 1 TRUE 39 1 0 0 1 1 0 FALSE 40 1 0 0 1 1 1 FALSE 41 1 0 1 0 0 0 TRUE 42 1 0 1 0 0 1 TRUE 43 1 0 1 0 1 0 TRUE 44 1 0 1 0 1 1 FALSE 45 1 0 1 1 0 0 FALSE 46 1 0 1 1 0 1 FALSE 47 1 0 1 1 1 0 FALSE 48 1 0 1 1 1 1 FALSE 49 1 1 0 0 0 0 FALSE 50 1 1 0 0 0 1 FALSE 51 1 1 0 0 1 0 FALSE 52 1 1 0 0 1 1 FALSE 53 1 1 0 1 0 0 FALSE 54 1 1 0 1 0 1 FALSE 55 1 1 0 1 1 0 FALSE 56 1 1 0 1 1 1 FALSE 57 1 1 1 0 0 0 FALSE 58 1 1 1 0 0 1 FALSE 59 1 1 1 0 1 0 FALSE 60 1 1 1 0 1 1 FALSE 61 1 1 1 1 0 0 FALSE 62 1 1 1 1 0 1 FALSE 63 1 1 1 1 1 0 FALSE 64 1 1 1 1 1 1 FALSE

Based on 6 flips there is 21 out of 64 for a probability of 32.8125%. With the progression we have we can find an equation or equation process to find the next sequence and then verify it by actually doing all the outcomes. The current progression is shown below.

 Flips No Back-to-Back Outcomes 2 3 3 5 4 8 5 13 6 21

If you know this sequence, you should be able to see it's a Fibonacci Sequence; where the previous 2 outcomes summed together become the next outcome total.

3 + 5 = 8

5 + 8 = 13

8 + 13 = 21

So, where does this lead? Well, by this understanding the next No Back-to-Back outcome count should be 13 + 21, or 34; you verify it in the table below.

 Flip # Outcome # 1 2 3 4 5 6 7 No Back-To-Back 1 0 0 0 0 0 0 0 TRUE 2 0 0 0 0 0 0 1 TRUE 3 0 0 0 0 0 1 0 TRUE 4 0 0 0 0 0 1 1 FALSE 5 0 0 0 0 1 0 0 TRUE 6 0 0 0 0 1 0 1 TRUE 7 0 0 0 0 1 1 0 FALSE 8 0 0 0 0 1 1 1 FALSE 9 0 0 0 1 0 0 0 TRUE 10 0 0 0 1 0 0 1 TRUE 11 0 0 0 1 0 1 0 TRUE 12 0 0 0 1 0 1 1 FALSE 13 0 0 0 1 1 0 0 FALSE 14 0 0 0 1 1 0 1 FALSE 15 0 0 0 1 1 1 0 FALSE 16 0 0 0 1 1 1 1 FALSE 17 0 0 1 0 0 0 0 TRUE 18 0 0 1 0 0 0 1 TRUE 19 0 0 1 0 0 1 0 TRUE 20 0 0 1 0 0 1 1 FALSE 21 0 0 1 0 1 0 0 TRUE 22 0 0 1 0 1 0 1 TRUE 23 0 0 1 0 1 1 0 FALSE 24 0 0 1 0 1 1 1 FALSE 25 0 0 1 1 0 0 0 FALSE 26 0 0 1 1 0 0 1 FALSE 27 0 0 1 1 0 1 0 FALSE 28 0 0 1 1 0 1 1 FALSE 29 0 0 1 1 1 0 0 FALSE 30 0 0 1 1 1 0 1 FALSE 31 0 0 1 1 1 1 0 FALSE 32 0 0 1 1 1 1 1 FALSE 33 0 1 0 0 0 0 0 TRUE 34 0 1 0 0 0 0 1 TRUE 35 0 1 0 0 0 1 0 TRUE 36 0 1 0 0 0 1 1 FALSE 37 0 1 0 0 1 0 0 TRUE 38 0 1 0 0 1 0 1 TRUE 39 0 1 0 0 1 1 0 FALSE 40 0 1 0 0 1 1 1 FALSE 41 0 1 0 1 0 0 0 TRUE 42 0 1 0 1 0 0 1 TRUE 43 0 1 0 1 0 1 0 TRUE 44 0 1 0 1 0 1 1 FALSE 45 0 1 0 1 1 0 0 FALSE 46 0 1 0 1 1 0 1 FALSE 47 0 1 0 1 1 1 0 FALSE 48 0 1 0 1 1 1 1 FALSE 49 0 1 1 0 0 0 0 FALSE 50 0 1 1 0 0 0 1 FALSE 51 0 1 1 0 0 1 0 FALSE 52 0 1 1 0 0 1 1 FALSE 53 0 1 1 0 1 0 0 FALSE 54 0 1 1 0 1 0 1 FALSE 55 0 1 1 0 1 1 0 FALSE 56 0 1 1 0 1 1 1 FALSE 57 0 1 1 1 0 0 0 FALSE 58 0 1 1 1 0 0 1 FALSE 59 0 1 1 1 0 1 0 FALSE 60 0 1 1 1 0 1 1 FALSE 61 0 1 1 1 1 0 0 FALSE 62 0 1 1 1 1 0 1 FALSE 63 0 1 1 1 1 1 0 FALSE 64 0 1 1 1 1 1 1 FALSE 65 1 0 0 0 0 0 0 TRUE 66 1 0 0 0 0 0 1 TRUE 67 1 0 0 0 0 1 0 TRUE 68 1 0 0 0 0 1 1 FALSE 69 1 0 0 0 1 0 0 TRUE 70 1 0 0 0 1 0 1 TRUE 71 1 0 0 0 1 1 0 FALSE 72 1 0 0 0 1 1 1 FALSE 73 1 0 0 1 0 0 0 TRUE 74 1 0 0 1 0 0 1 TRUE 75 1 0 0 1 0 1 0 TRUE 76 1 0 0 1 0 1 1 FALSE 77 1 0 0 1 1 0 0 FALSE 78 1 0 0 1 1 0 1 FALSE 79 1 0 0 1 1 1 0 FALSE 80 1 0 0 1 1 1 1 FALSE 81 1 0 1 0 0 0 0 TRUE 82 1 0 1 0 0 0 1 TRUE 83 1 0 1 0 0 1 0 TRUE 84 1 0 1 0 0 1 1 FALSE 85 1 0 1 0 1 0 0 TRUE 86 1 0 1 0 1 0 1 TRUE 87 1 0 1 0 1 1 0 FALSE 88 1 0 1 0 1 1 1 FALSE 89 1 0 1 1 0 0 0 FALSE 90 1 0 1 1 0 0 1 FALSE 91 1 0 1 1 0 1 0 FALSE 92 1 0 1 1 0 1 1 FALSE 93 1 0 1 1 1 0 0 FALSE 94 1 0 1 1 1 0 1 FALSE 95 1 0 1 1 1 1 0 FALSE 96 1 0 1 1 1 1 1 FALSE 97 1 1 0 0 0 0 0 FALSE 98 1 1 0 0 0 0 1 FALSE 99 1 1 0 0 0 1 0 FALSE 100 1 1 0 0 0 1 1 FALSE 101 1 1 0 0 1 0 0 FALSE 102 1 1 0 0 1 0 1 FALSE 103 1 1 0 0 1 1 0 FALSE 104 1 1 0 0 1 1 1 FALSE 105 1 1 0 1 0 0 0 FALSE 106 1 1 0 1 0 0 1 FALSE 107 1 1 0 1 0 1 0 FALSE 108 1 1 0 1 0 1 1 FALSE 109 1 1 0 1 1 0 0 FALSE 110 1 1 0 1 1 0 1 FALSE 111 1 1 0 1 1 1 0 FALSE 112 1 1 0 1 1 1 1 FALSE 113 1 1 1 0 0 0 0 FALSE 114 1 1 1 0 0 0 1 FALSE 115 1 1 1 0 0 1 0 FALSE 116 1 1 1 0 0 1 1 FALSE 117 1 1 1 0 1 0 0 FALSE 118 1 1 1 0 1 0 1 FALSE 119 1 1 1 0 1 1 0 FALSE 120 1 1 1 0 1 1 1 FALSE 121 1 1 1 1 0 0 0 FALSE 122 1 1 1 1 0 0 1 FALSE 123 1 1 1 1 0 1 0 FALSE 124 1 1 1 1 0 1 1 FALSE 125 1 1 1 1 1 0 0 FALSE 126 1 1 1 1 1 0 1 FALSE 127 1 1 1 1 1 1 0 FALSE 128 1 1 1 1 1 1 1 FALSE

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Jehocifer

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 Posted: October 2, 2009, 4:29 pm - IP Logged

"Now, the question is going to have to be ...... What is the probability of tossing a coin 36 times without getting 2 Tails in a row?    At least for now....thats the question. LOL "

It really doesn't matter: Heads They Win, Tails You Lose! LOL!!!

A mind once stretched by a new idea never returns to its original dimensions!

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 Posted: October 3, 2009, 1:08 pm - IP Logged

"Where you went wrong is not considering ALL possibilities."

I was dismissing some of the possibilities, albeit indirectly, and it took a while to find the fault in my logic.

I'm glad to see that the number of  outcomes with no back to back flips follows the Fibonacci sequence. If it wasn't for that saving grace I'd be completely dismayed at seeing  the calculations. Most math is fairly elegant, and most probability follows a fairly direct path of factorials or squares, and simple arithmetic. Even calculating as the remainder of  the probability of the opposite outcome seems perfectly simple and straighforward. For this exercise I'm still not seeing anything simple and direct in calculating the probabilities, and I find it extremely counterintuitive that the differences between the probability at n flips and n+1 don't change by the same factor.

Any chance the excel formual you used can essily do a list for not having any 3 in a row occcurrences? I'd be curious to see if there's an part of that that has a simple pattern.

The Quantum Master
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 Posted: October 3, 2009, 2:28 pm - IP Logged

"Where you went wrong is not considering ALL possibilities."

I was dismissing some of the possibilities, albeit indirectly, and it took a while to find the fault in my logic.

I'm glad to see that the number of  outcomes with no back to back flips follows the Fibonacci sequence. If it wasn't for that saving grace I'd be completely dismayed at seeing  the calculations. Most math is fairly elegant, and most probability follows a fairly direct path of factorials or squares, and simple arithmetic. Even calculating as the remainder of  the probability of the opposite outcome seems perfectly simple and straighforward. For this exercise I'm still not seeing anything simple and direct in calculating the probabilities, and I find it extremely counterintuitive that the differences between the probability at n flips and n+1 don't change by the same factor.

Any chance the excel formual you used can essily do a list for not having any 3 in a row occcurrences? I'd be curious to see if there's an part of that that has a simple pattern.

It's just a matter of setting up the proper testing IF statements; you can use the same set of outcomes and reform the testing IF condition. To understand this correctly, you are looking for a condition when there are no 1's in a triple consecutive sequence, i.e. ( 1 1 1 )?

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Jehocifer

The Quantum Master
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 Posted: October 3, 2009, 4:13 pm - IP Logged

Well, OK... here is the expansion and more than just a 3 non-consecutive flips. I've expanded it to 5 non-consecutive flips. Below is a table with just the counts. You can figure the Probability by taking the count and dividing by the value, 2n, where n is the number of flips; to get the percentage, multiply by 100. Also, I have found that the counts are related to the number of non-consecutives in determining the progression sequence. It's a play on the basic idea of the Fibonacci Sequence by expanding the sum from the last 2 values to increase the sum set by one more value. The non-consecutive count for 3 is the sum of the last 3 values; the non-consecutive count for 4 is the sum of the last 4 values and so on. The numbers highlighted in each column are the starting Fibonacci numbers.

 No Back-to-Back Counts Non-Consecutive Flips Threshold-> 2 3 4 5 Fibonacci Sum Pattern -> fn-1 + fn-2 = fn fn-1 + fn-2 + fn-3 = fn fn-1 + fn-2 + fn-3 + fn-4 = fn fn-1 + fn-2 + fn-3 + fn-4 + fn-5 = fn n Number of Flips 2 3 - - - 3 5 7 - - 4 8 13 15 - 5 13 24 29 31 6 21 44 56 61 7 34 81 108 120 8 55 149 208 236 9 89 274 401 464 10 144 504 773 912 11 233 927 1490 1793 12 377 1705 2872 3525 13 610 3136 5536 6930 14 987 5768 10671 13624 15 1597 10609 20569 26784 16 2584 19513 39648 52656 17 4181 35890 76424 103519 18 6765 66012 147312 203513 19 10946 121415 283953 400096 20 17711 223317 547337 786568 21 28657 410744 1055026 1546352 22 46368 755476 2033628 3040048 23 75025 1389537 3919944 5976577 24 121393 2555757 7555935 11749641 25 196418 4700770 14564533 23099186 26 317811 8646064 28074040 45411804 27 514229 15902591 54114452 89277256 28 832040 29249425 104308960 175514464 29 1346269 53798080 201061985 345052351 30 2178309 98950096 387559437 678355061 31 3524578 181997601 747044834 1333610936 32 5702887 334745777 1439975216 2621810068 33 9227465 615693474 2775641472 5154342880 34 14930352 1132436852 5350220959 10133171296 35 24157817 2082876103 10312882481 19921290241 36 39088169 3831006429 19878720128 39164225421 37 63245986 7046319384 38317465040 76994839906 38 102334155 12960201916 73859288608 151367869744 39 165580141 23837527729 142368356257 297581396608 40 267914296 43844049029 274423830033 585029621920 41 433494437 80641778674 528968939938 1150137953599 42 701408733 148323355432 1019620414836 2261111681777 43 1134903170 272809183135 1965381541064 4445228523648 44 1836311903 501774317241 3788394725871 8739089177552 45 2971215073 922906855808 7302365621709 17180596958496 46 4807526976 1697490356184 14075762303480 33776164295072 47 7778742049 3122171529233 27131904192124 66402190636545 48 12586269025 5742568741225 52298426843184 130543269591313 49 20365011074 10562230626642 100808458960497 256641310658978 50 32951280099 19426970897100 194314552299285 504543532140404

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Jehocifer

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 Posted: October 4, 2009, 12:09 am - IP Logged

@ KY Floyd

Not that I ever needed those particular questions answered but glad you guys are around for all the statistics/probabilities/math!  Thank you.

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 Posted: October 4, 2009, 10:35 am - IP Logged

@ KY Floyd