What is the probability of this happening again?Prev Topic Next Topic

WIN D

Stone Mountain*Georgia
United States
Member #828
November 2, 2002
10,491 Posts
Offline

Sep 30, 2009, 12:06 pm

   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
KY Floyd

NY
United States
Member #23,834
October 16, 2005
4,772 Posts
Offline

Oct 1, 2009, 2:22 am

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.
WIN D

Stone Mountain*Georgia
United States
Member #828
November 2, 2002
10,491 Posts
Offline

Oct 1, 2009, 4:27 pm

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
WIN D

Stone Mountain*Georgia
United States
Member #828
November 2, 2002
10,491 Posts
Offline

Oct 1, 2009, 4:51 pm

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
WIN D

Stone Mountain*Georgia
United States
Member #828
November 2, 2002
10,491 Posts
Offline

Oct 1, 2009, 8:05 pm

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
KY Floyd

NY
United States
Member #23,834
October 16, 2005
4,772 Posts
Offline

Oct 2, 2009, 1:16 am

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.

Native American Eagle Sun

MN
United States
Member #21
December 7, 2001
4,811 Posts
Offline

Oct 2, 2009, 10:14 am

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%

The One Over None
I Know...

KY Floyd

NY
United States
Member #23,834
October 16, 2005
4,772 Posts
Offline

Oct 2, 2009, 2:56 pm

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.

JADELottery

Native American Eagle Sun

MN
United States
Member #21
December 7, 2001
4,811 Posts
Offline

Oct 2, 2009, 4:24 pm

Quote: Originally posted by KY Floyd on Oct 2, 2009

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

The One Over None
I Know...

Raven62

25

New Jersey
United States
Member #17,842
June 28, 2005
180,983 Posts
Offline

Oct 2, 2009, 4:29 pm

"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!

Catch-22: A dilemma or difficult circumstance from which there is no escape because of mutually conflicting or dependent conditions.

Corruptissima re publica plurimae leges: When the republic is at its most corrupt the laws are most numerous.

The best way to learn is to never stop being an Experiential Student!
KY Floyd

NY
United States
Member #23,834
October 16, 2005
4,772 Posts
Offline

Oct 3, 2009, 1:08 pm

"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.
JADELottery

Native American Eagle Sun

MN
United States
Member #21
December 7, 2001
4,811 Posts
Offline

Oct 3, 2009, 2:28 pm

Quote: Originally posted by KY Floyd on Oct 3, 2009
"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 )?

The One Over None
I Know...

JADELottery

Native American Eagle Sun

MN
United States
Member #21
December 7, 2001
4,811 Posts
Offline

Oct 3, 2009, 4:13 pm

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 ->	f_n-1 + f_n-2 = f_n	f_n-1 + f_n-2 + f_n-3 = f_n	f_n-1 + f_n-2 + f_n-3 + f_n-4 = f_n	f_n-1 + f_n-2 + f_n-3 + f_n-4 + f_n-5 = f_n
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

The One Over None
I Know...

truecritic

Michigan
United States
Member #22,394
September 24, 2005
1,583 Posts
Offline

Oct 4, 2009, 12:09 am

@ KY Floyd

@ JADELottery

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

COLUMBUS,GA.
United States
Member #4,924
June 3, 2004
6,719 Posts
Offline

Oct 4, 2009, 10:35 am

Quote: Originally posted by truecritic on Oct 4, 2009
@ KY Floyd

@ JADELottery

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

How right you are! It's nice to know, for questions like these, we have brilliant minds that will come forward with the solution. Thanks to both of them.

New Topic New Poll

Subscribe to this topic

What is the probability of this happening again?Prev TopicNext Topic

What is the probability of this happening again?Prev Topic Next Topic