Este post é uma reedição de um post meu no euromilhoes.com.
Antes de se avançar para qualquer estudo estatístico em totolotarias, é conveniente termos a noção de qual é a probabilidade de acertarmos com um dado conjunto de números. Só assim é possível aferir a fiabilidade de um conjunto de filtros, ou uma metodologia de escolha de números e/ou filtros, por via da sua aplicação ao historial do concurso. É claro para toda a gente que, se uma dada metodologia tem resultados inferiores ou sensivelmente iguais à probabilidade natural dum conjunto de números de igual tamanho, essa metodologia está desafinada ou é uma treta (ou ambas).
Por esse motivo, fiz este estudo em Novembro de 2006 para o Euromilhões (números e estrelas) e para o totoloto. A primeira coluna das tabelas é a quantidade de números escolhida e a primeira fila é a quantidade de acertos. Nas tabelas onde essa primeira fila é 1, 2, 3, etc., a probabilidade é calculada para exactamente esses acertos, nem mais, nem menos; Nas tabelas onde a primeira fila é 1+, 2+, 3+, etc., a probabilidade é calculada para pelo menos esses acertos (1 ou mais, 2 ou mais, etc.).
A probabilidade de se acertar x elementos num total de n de um conjunto de tamanho N, em sorteios de K elementos é calculada recorrendo à fórmula da distribuição hipergeométrica. O Excel, amigo de todas as horas nos meandros matemáticos, tem uma expressão directa para o calcular, HYPGEOMDIST(x,n,K,N). A fórmula completa, usando combinações, é a seguinte:
A título de exemplo, para o caso do euromilhões, querendo acertar em 5 escolhendo 20, temos que x = 5, N = 50, K = 5 e n = 20:
0 | 1 | 2 | 3 | 4 | 5 | |
5 | 57,66387% | 35,16090% | 6,69731% | 0,46725% | 0,01062% | 0,00005% |
6 | 51,25677% | 38,44258% | 9,37624% | 0,89298% | 0,03115% | 0,00028% |
7 | 45,43214% | 40,77243% | 12,23173% | 1,49167% | 0,07103% | 0,00099% |
8 | 40,14933% | 42,26246% | 15,17114% | 2,27567% | 0,13876% | 0,00264% |
9 | 35,36965% | 43,01714% | 18,11248% | 3,25096% | 0,24382% | 0,00595% |
10 | 31,05628% | 43,13372% | 20,98397% | 4,41768% | 0,39646% | 0,01189% |
11 | 27,17424% | 42,70238% | 23,72355% | 5,77059% | 0,60743% | 0,02181% |
12 | 23,69037% | 41,80653% | 26,27839% | 7,29955% | 0,88778% | 0,03738% |
13 | 20,57321% | 40,52299% | 28,60447% | 8,98998% | 1,24861% | 0,06074% |
14 | 17,79305% | 38,92229% | 30,66605% | 10,82331% | 1,70081% | 0,09449% |
15 | 15,32179% | 37,06885% | 32,43525% | 12,77752% | 2,25486% | 0,14173% |
16 | 13,13296% | 35,02124% | 33,89152% | 14,82754% | 2,92058% | 0,20616% |
17 | 11,20165% | 32,83241% | 35,02124% | 16,94576% | 3,70689% | 0,29206% |
18 | 9,50443% | 30,54994% | 35,81718% | 19,10249% | 4,62157% | 0,40439% |
19 | 8,01936% | 28,21627% | 36,27806% | 21,26645% | 5,67105% | 0,54881% |
20 | 6,72592% | 25,86890% | 36,40809% | 23,40520% | 6,86014% | 0,73175% |
21 | 5,60493% | 23,54070% | 36,21647% | 25,48566% | 8,19182% | 0,96042% |
22 | 4,63856% | 21,26008% | 35,71693% | 27,47456% | 9,66698% | 1,24290% |
23 | 3,81025% | 19,05124% | 34,92727% | 29,33891% | 11,28419% | 1,58815% |
24 | 3,10465% | 16,93443% | 33,86887% | 31,04646% | 13,03951% | 2,00608% |
25 | 2,50760% | 14,92618% | 32,56622% | 32,56622% | 14,92618% | 2,50760% |
26 | 2,00608% | 13,03951% | 31,04646% | 33,86887% | 16,93443% | 3,10465% |
27 | 1,58815% | 11,28419% | 29,33891% | 34,92727% | 19,05124% | 3,81025% |
28 | 1,24290% | 9,66698% | 27,47456% | 35,71693% | 21,26008% | 4,63856% |
29 | 0,96042% | 8,19182% | 25,48566% | 36,21647% | 23,54070% | 5,60493% |
30 | 0,73175% | 6,86014% | 23,40520% | 36,40809% | 25,86890% | 6,72592% |
31 | 0,54881% | 5,67105% | 21,26645% | 36,27806% | 28,21627% | 8,01936% |
32 | 0,40439% | 4,62157% | 19,10249% | 35,81718% | 30,54994% | 9,50443% |
33 | 0,29206% | 3,70689% | 16,94576% | 35,02124% | 32,83241% | 11,20165% |
34 | 0,20616% | 2,92058% | 14,82754% | 33,89152% | 35,02124% | 13,13296% |
35 | 0,14173% | 2,25486% | 12,77752% | 32,43525% | 37,06885% | 15,32179% |
36 | 0,09449% | 1,70081% | 10,82331% | 30,66605% | 38,92229% | 17,79305% |
37 | 0,06074% | 1,24861% | 8,98998% | 28,60447% | 40,52299% | 20,57321% |
38 | 0,03738% | 0,88778% | 7,29955% | 26,27839% | 41,80653% | 23,69037% |
39 | 0,02181% | 0,60743% | 5,77059% | 23,72355% | 42,70238% | 27,17424% |
40 | 0,01189% | 0,39646% | 4,41768% | 20,98397% | 43,13372% | 31,05628% |
41 | 0,00595% | 0,24382% | 3,25096% | 18,11248% | 43,01714% | 35,36965% |
42 | 0,00264% | 0,13876% | 2,27567% | 15,17114% | 42,26246% | 40,14933% |
43 | 0,00099% | 0,07103% | 1,49167% | 12,23173% | 40,77243% | 45,43214% |
44 | 0,00028% | 0,03115% | 0,89298% | 9,37624% | 38,44258% | 51,25677% |
45 | 0,00005% | 0,01062% | 0,46725% | 6,69731% | 35,16090% | 57,66387% |
46 | 0,00000% | 0,00217% | 0,19540% | 4,29874% | 30,80764% | 64,69605% |
47 | 0,00000% | 0,00000% | 0,05102% | 2,29592% | 25,25510% | 72,39796% |
48 | 0,00000% | 0,00000% | 0,00000% | 0,81633% | 18,36735% | 80,81633% |
49 | 0,00000% | 0,00000% | 0,00000% | 0,00000% | 10,00000% | 90,00000% |
50 | 0,00000% | 0,00000% | 0,00000% | 0,00000% | 0,00000% | 100,00000% |
1+ | 2+ | 3+ | 4+ | |
5 | 42,33613% | 7,17523% | 0,47792% | 0,01067% |
6 | 48,74323% | 10,30065% | 0,92441% | 0,03143% |
7 | 54,56786% | 13,79543% | 1,56370% | 0,07202% |
8 | 59,85067% | 17,58821% | 2,41707% | 0,14140% |
9 | 64,63035% | 21,61321% | 3,50073% | 0,24977% |
10 | 68,94372% | 25,81000% | 4,82603% | 0,40835% |
11 | 72,82576% | 30,12337% | 6,39983% | 0,62924% |
12 | 76,30963% | 34,50311% | 8,22472% | 0,92516% |
13 | 79,42679% | 38,90379% | 10,29933% | 1,30935% |
14 | 82,20695% | 43,28466% | 12,61861% | 1,79530% |
15 | 84,67821% | 47,60936% | 15,17411% | 2,39659% |
16 | 86,86704% | 51,84580% | 17,95428% | 3,12673% |
17 | 88,79835% | 55,96594% | 20,94470% | 3,99894% |
18 | 90,49557% | 59,94563% | 24,12845% | 5,02596% |
19 | 91,98064% | 63,76437% | 27,48631% | 6,21986% |
20 | 93,27408% | 67,40518% | 30,99709% | 7,59189% |
21 | 94,39507% | 70,85437% | 34,63790% | 9,15224% |
22 | 95,36144% | 74,10136% | 38,38443% | 10,90987% |
23 | 96,18975% | 77,13851% | 42,21125% | 12,87234% |
24 | 96,89535% | 79,96092% | 46,09205% | 15,04559% |
25 | 97,49240% | 82,56622% | 50,00000% | 17,43378% |
26 | 97,99392% | 84,95441% | 53,90795% | 20,03908% |
27 | 98,41185% | 87,12766% | 57,78875% | 22,86149% |
28 | 98,75710% | 89,09013% | 61,61557% | 25,89864% |
29 | 99,03958% | 90,84776% | 65,36210% | 29,14563% |
30 | 99,26825% | 92,40811% | 69,00291% | 32,59482% |
31 | 99,45119% | 93,78014% | 72,51369% | 36,23563% |
32 | 99,59561% | 94,97404% | 75,87155% | 40,05437% |
33 | 99,70794% | 96,00106% | 79,05530% | 44,03406% |
34 | 99,79384% | 96,87327% | 82,04572% | 48,15420% |
35 | 99,85827% | 97,60341% | 84,82589% | 52,39064% |
36 | 99,90551% | 98,20470% | 87,38139% | 56,71534% |
37 | 99,93926% | 98,69065% | 89,70067% | 61,09621% |
38 | 99,96262% | 99,07484% | 91,77528% | 65,49689% |
39 | 99,97819% | 99,37076% | 93,60017% | 69,87663% |
40 | 99,98811% | 99,59165% | 95,17397% | 74,19000% |
41 | 99,99405% | 99,75023% | 96,49927% | 78,38679% |
42 | 99,99736% | 99,85860% | 97,58293% | 82,41179% |
43 | 99,99901% | 99,92798% | 98,43630% | 86,20457% |
44 | 99,99972% | 99,96857% | 99,07559% | 89,69935% |
45 | 99,99995% | 99,98933% | 99,52208% | 92,82477% |
46 | 100,00000% | 99,99783% | 99,80243% | 95,50369% |
47 | 100,00000% | 100,00000% | 99,94898% | 97,65306% |
48 | 100,00000% | 100,00000% | 100,00000% | 99,18367% |
49 | 100,00000% | 100,00000% | 100,00000% | 100,00000% |
50 | 100,00000% | 100,00000% | 100,00000% | 100,00000% |
0 | 1 | 2 | |
2 | 58,33% | 38,89% | 2,78% |
3 | 41,67% | 50,00% | 8,33% |
4 | 27,78% | 55,56% | 16,67% |
5 | 16,67% | 55,56% | 27,78% |
6 | 8,33% | 50,00% | 41,67% |
7 | 2,78% | 38,89% | 58,33% |
8 | 0,00% | 22,22% | 77,78% |
9 | 0,00% | 0,00% | 100,00% |
1+ | |
3 | 58,33% |
4 | 72,22% |
5 | 83,33% |
6 | 91,67% |
7 | 97,22% |
8 | 100,00% |
9 | 100,00% |
0 | 1 | 2 | 3 | 4 | 5 | 6 | |
6 | 43,59650% | 41,30195% | 13,23780% | 1,76504% | 0,09686% | 0,00184% | 0,00001% |
7 | 37,51327% | 42,58263% | 16,80893% | 2,87332% | 0,21550% | 0,00631% | 0,00005% |
8 | 32,15423% | 42,87230% | 20,27744% | 4,26893% | 0,41047% | 0,01642% | 0,00020% |
9 | 27,44873% | 42,34947% | 23,52748% | 5,93486% | 0,70281% | 0,03604% | 0,00060% |
10 | 23,33142% | 41,17310% | 26,46842% | 7,84249% | 1,11279% | 0,07028% | 0,00150% |
11 | 19,74197% | 39,48394% | 29,03231% | 9,95394% | 1,65899% | 0,12555% | 0,00330% |
12 | 16,62482% | 37,40584% | 31,17153% | 12,22413% | 2,35751% | 0,20956% | 0,00661% |
13 | 13,92890% | 35,04691% | 32,85648% | 14,60288% | 3,22122% | 0,33133% | 0,01227% |
14 | 11,60742% | 32,50077% | 34,07339% | 17,03669% | 4,25917% | 0,50108% | 0,02147% |
15 | 9,61758% | 29,84765% | 34,82225% | 19,47051% | 5,47608% | 0,73014% | 0,03579% |
16 | 7,92036% | 27,15551% | 35,11488% | 21,84926% | 6,87194% | 1,03079% | 0,05727% |
17 | 6,48029% | 24,48110% | 34,97300% | 24,11931% | 8,44176% | 1,41604% | 0,08850% |
18 | 5,26524% | 21,87098% | 34,42655% | 26,22975% | 10,17533% | 1,89940% | 0,13275% |
19 | 4,24616% | 19,36248% | 33,51199% | 28,13352% | 12,05722% | 2,49460% | 0,19402% |
20 | 3,39693% | 16,98463% | 32,27081% | 29,78844% | 14,06676% | 3,21526% | 0,27718% |
21 | 2,69411% | 14,75906% | 30,74804% | 31,15802% | 16,17820% | 4,07451% | 0,38805% |
22 | 2,11680% | 12,70083% | 28,99101% | 32,21224% | 18,36098% | 5,08458% | 0,53357% |
23 | 1,64640% | 10,81922% | 27,04805% | 32,92806% | 20,58004% | 6,25633% | 0,72188% |
24 | 1,26646% | 9,11854% | 24,96743% | 33,28991% | 22,79635% | 7,59878% | 0,96251% |
25 | 0,96251% | 7,59878% | 22,79635% | 33,28991% | 24,96743% | 9,11854% | 1,26646% |
26 | 0,72188% | 6,25633% | 20,58004% | 32,92806% | 27,04805% | 10,81922% | 1,64640% |
27 | 0,53357% | 5,08458% | 18,36098% | 32,21224% | 28,99101% | 12,70083% | 2,11680% |
28 | 0,38805% | 4,07451% | 16,17820% | 31,15802% | 30,74804% | 14,75906% | 2,69411% |
29 | 0,27718% | 3,21526% | 14,06676% | 29,78844% | 32,27081% | 16,98463% | 3,39693% |
30 | 0,19402% | 2,49460% | 12,05722% | 28,13352% | 33,51199% | 19,36248% | 4,24616% |
31 | 0,13275% | 1,89940% | 10,17533% | 26,22975% | 34,42655% | 21,87098% | 5,26524% |
32 | 0,08850% | 1,41604% | 8,44176% | 24,11931% | 34,97300% | 24,48110% | 6,48029% |
33 | 0,05727% | 1,03079% | 6,87194% | 21,84926% | 35,11488% | 27,15551% | 7,92036% |
34 | 0,03579% | 0,73014% | 5,47608% | 19,47051% | 34,82225% | 29,84765% | 9,61758% |
35 | 0,02147% | 0,50108% | 4,25917% | 17,03669% | 34,07339% | 32,50077% | 11,60742% |
36 | 0,01227% | 0,33133% | 3,22122% | 14,60288% | 32,85648% | 35,04691% | 13,92890% |
37 | 0,00661% | 0,20956% | 2,35751% | 12,22413% | 31,17153% | 37,40584% | 16,62482% |
38 | 0,00330% | 0,12555% | 1,65899% | 9,95394% | 29,03231% | 39,48394% | 19,74197% |
39 | 0,00150% | 0,07028% | 1,11279% | 7,84249% | 26,46842% | 41,17310% | 23,33142% |
40 | 0,00060% | 0,03604% | 0,70281% | 5,93486% | 23,52748% | 42,34947% | 27,44873% |
41 | 0,00020% | 0,01642% | 0,41047% | 4,26893% | 20,27744% | 42,87230% | 32,15423% |
42 | 0,00005% | 0,00631% | 0,21550% | 2,87332% | 16,80893% | 42,58263% | 37,51327% |
43 | 0,00001% | 0,00184% | 0,09686% | 1,76504% | 13,23780% | 41,30195% | 43,59650% |
44 | 0,00000% | 0,00031% | 0,03382% | 0,94709% | 9,70772% | 38,83089% | 50,48016% |
45 | 0,00000% | 0,00000% | 0,00708% | 0,40590% | 6,39289% | 34,94780% | 58,24633% |
46 | 0,00000% | 0,00000% | 0,00000% | 0,10855% | 3,50087% | 29,40729% | 66,98328% |
47 | 0,00000% | 0,00000% | 0,00000% | 0,00000% | 1,27551% | 21,93878% | 76,78571% |
48 | 0,00000% | 0,00000% | 0,00000% | 0,00000% | 0,00000% | 12,24490% | 87,75510% |
49 | 0,00000% | 0,00000% | 0,00000% | 0,00000% | 0,00000% | 0,00000% | 100,00000% |
1+ | 2+ | 3+ | 4+ | 5+ | |
7 | 56,40350% | 15,10156% | 1,86375% | 0,09871% | 0,00185% |
8 | 62,48673% | 19,90411% | 3,09518% | 0,22186% | 0,00636% |
9 | 67,84577% | 24,97347% | 4,69603% | 0,42709% | 0,01662% |
10 | 72,55127% | 30,20180% | 6,67432% | 0,73945% | 0,03664% |
11 | 76,66858% | 35,49548% | 9,02706% | 1,18457% | 0,07178% |
12 | 80,25803% | 40,77408% | 11,74177% | 1,78784% | 0,12885% |
13 | 83,37518% | 45,96934% | 14,79781% | 2,57368% | 0,21616% |
14 | 86,07110% | 51,02418% | 18,16770% | 3,56482% | 0,34360% |
15 | 88,39258% | 55,89181% | 21,81842% | 4,78173% | 0,52255% |
16 | 90,38242% | 60,53478% | 25,71252% | 6,24202% | 0,76594% |
17 | 92,07964% | 64,92414% | 29,80926% | 7,96000% | 1,08806% |
18 | 93,51971% | 69,03861% | 34,06561% | 9,94630% | 1,50454% |
19 | 94,73476% | 72,86378% | 38,43723% | 12,20748% | 2,03215% |
20 | 95,75384% | 76,39136% | 42,87937% | 14,74585% | 2,68862% |
21 | 96,60307% | 79,61844% | 47,34763% | 17,55920% | 3,49244% |
22 | 97,30589% | 82,54682% | 51,79878% | 20,64076% | 4,46256% |
23 | 97,88320% | 85,18237% | 56,19136% | 23,97912% | 5,61814% |
24 | 98,35360% | 87,53438% | 60,48632% | 27,55826% | 6,97822% |
25 | 98,73354% | 89,61499% | 64,64756% | 31,35765% | 8,56130% |
26 | 99,03749% | 91,43870% | 68,64235% | 35,35244% | 10,38501% |
27 | 99,27812% | 93,02178% | 72,44174% | 39,51368% | 12,46562% |
28 | 99,46643% | 94,38186% | 76,02088% | 43,80864% | 14,81763% |
29 | 99,61195% | 95,53744% | 79,35924% | 48,20122% | 17,45318% |
30 | 99,72282% | 96,50756% | 82,44080% | 52,65237% | 20,38156% |
31 | 99,80598% | 97,31138% | 85,25415% | 57,12063% | 23,60864% |
32 | 99,86725% | 97,96785% | 87,79252% | 61,56277% | 27,13622% |
33 | 99,91150% | 98,49546% | 90,05370% | 65,93439% | 30,96139% |
34 | 99,94273% | 98,91194% | 92,04000% | 70,19074% | 35,07586% |
35 | 99,96421% | 99,23406% | 93,75798% | 74,28748% | 39,46522% |
36 | 99,97853% | 99,47745% | 95,21827% | 78,18158% | 44,10819% |
37 | 99,98773% | 99,65640% | 96,43518% | 81,83230% | 48,97582% |
38 | 99,99339% | 99,78384% | 97,42632% | 85,20219% | 54,03066% |
39 | 99,99670% | 99,87115% | 98,21216% | 88,25823% | 59,22592% |
40 | 99,99850% | 99,92822% | 98,81543% | 90,97294% | 64,50452% |
41 | 99,99940% | 99,96336% | 99,26055% | 93,32568% | 69,79820% |
42 | 99,99980% | 99,98338% | 99,57291% | 95,30397% | 75,02653% |
43 | 99,99995% | 99,99364% | 99,77814% | 96,90482% | 80,09589% |
44 | 99,99999% | 99,99815% | 99,90129% | 98,13625% | 84,89844% |
45 | 100,00000% | 99,99969% | 99,96586% | 99,01877% | 89,31104% |
46 | 100,00000% | 100,00000% | 99,99292% | 99,58702% | 93,19413% |
47 | 100,00000% | 100,00000% | 100,00000% | 99,89145% | 96,39058% |
48 | 100,00000% | 100,00000% | 100,00000% | 100,00000% | 98,72449% |
49 | 100,00000% | 100,00000% | 100,00000% | 100,00000% | 100,00000% |
Abraços!