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:

 formula_hypgeomdist

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:

formula_hypgeomdist_5_20_5_50

Euromilhões – Números

  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%

Probabilidades de acertos exactos no euromilhões

  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%

Probabilidades de acertos 'pelo menos' no euromilhões 

Euromilhões – Estrelas

  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%

Probabilidades de acertos exactos no euromilhões (estrelas)

  1+
3 58,33%
4 72,22%
5 83,33%
6 91,67%
7 97,22%
8 100,00%
9 100,00%

Probabilidades de acertos 'pelo menos' no euromilhões (estrelas)

Totoloto

  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%

Probabilidades de acertos exactos no totoloto

  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%

Probabilidades de acertos 'pelo menos' no totoloto

Abraços!

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