Probabilidades de acerto no euromilhões e totoloto pela quantidade de números apostada

Início » Blog » Probabilidades de acerto no euromilhões e totoloto pela quantidade de números apostada

14 2009 Outubro 01:15

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%

	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%

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

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%

	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%