Tackling the 6/49 Lottery and Debunking Common Myths with Probabilistic Methods and Combinatorial Designs

1/16 Tackling the 6/49 Lottery and D ebunking Com mon Myth s with Probabil istic Method s and Combinat orial Design s Ralph Stömmer * * Private Researcher, Karl-Birzer-Str. 20, 85521 Ottobrunn, Germany e-mail: ralph.stoemmer@trajectorix.de (Working paper, version 1, March 21, 2026) Abstract At the end, the hous e always win s! This simp le truth hold s for all public games of chance. Nevertheless, since lotteries have existed, people have tried everything to give luck a helping hand. Th is article com pares objective scientific approaches to t ac kle the 6/49 lotte r y: probabil istic methods and combinatorial desi gns. Th e mathematical models developed herein ca n be modified and applied to other lotte ries. Additionally, this work introd uces the newly const ructed (49, 6, 5) covering design, which meets the Schönheim bound. For lottery designs and for covering designs, a ben chmark b as ed on probabilistic methods is pre se nted. It is dem onstrated that common attempts to outwit the odds correspond to limitations of numbers to subsets, which disproportionate l y reduce the cha nces of winning. Keywords: lottery ; combinatoria l design; covering des ign ; lottery des i gn; probability t heory Mathematics s ubject cla ss ification codes: 05B05 ; 05B0 7 ; 05B30; 05B4 0; 62K05; 62K 10; 62E17 2/16 1. Introduction There are many lotteries around the world. A common lottery scheme is (49, 6, 6, t), known as 6/49 lottery, where 6 numbers ar e randomly drawn by the lottery agency from 49 numbers without replacement. Beforehand, p eople buy tickets and choose 6 numbers per ticket. People achi eve a t-hit if the 6 numbers drawn intersect the 6 numbers on their ticket in t common elements. The prize depends on the c ountry, the lottery agency, and the rules o f the game. In general, the prize s cales with t. F or some lotteries the prize starts at t = 3, sometimes sufficient to cover the ticket costs, up to t = 6 for the jackpot. There a re     = 13,983,816 possi ble combinations to choose 6 from 49 numbers. From that large quantity, the jackpot is just a s ingl e combination. The calculation of it s probability P(6) is intuitively clear: P(6) = 1 / 13,983,816 = 0.0000072%. Despite the extremely low value, a lot of people try their luck, on occasion on a weekly basis. As suming that 13 million players would particip ate at a na tionwide draw, one should expect 13 million x P(6) ≈ 1 winner on average. I f the lucky on e tells ne ws media that he produced the winning combination from birthdays, questionable attributions can backfire. There is a more sophisticated evaluation of odds. Th is introductory example might serve as a warm-up exercise to slowly deepen the understa nding of complementary probabilities. Complementary probabilities are crucial in de termining probabilities with not just one, but with many lottery ticke ts . It turns out that probabilities for singl e tickets are we ll known, but work published on many tickets is rare. Let P (6) again denote the probability to achieve a 6-hit, the jackpot, with a single ticket. The complementary probability, to achieve no 6-hit, calculates to [1 – P(6)] = 99.99999%. Ending up empty- handed with a single ti cket is nothing new. Let´s consider the unrealistic case that we were able to submit v = 5 mi llion tickets, filled out at random. To achieve no 6-hit at all 3/16 with 5 million ti ckets calculates to [1 – P (6)] 5million ≈ 70%. Again, we take the complementary v alue to obtain the probability to win the jackpot with v = 5 million tickets: P(jackpot, v = 5 million ) = 1 – [1 – P(6)] 5million ≈ 30%. That is a quite reasonable chance of winning, if bulk submission of tickets was still allowed and technically feasible. However, such loophole s were closed d ecades a go, after syndicates were successfully tackling lot teries with ticket quantities and logi stic syst ems beyond the capabilities of individuals [1]. At the end, the house always wins. What is left for the pl ayer? The choice of the numbers on each ticket, and t he number of tickets, up to a maximum qua nti ty limited by the lottery agency. However, there is still some room for maneuver . The next chapter reveals the sc ientific wo rld hidden in between these two options. 2. Theory and application 2.1. Tackling the 6/49 lottery with probabilistic methods Although the basics of probabilistic methods are widely known, some insights have not been widely disseminated. The probability P(t) to achieve a t -hit is given by the fraction P(t) = M t /N, the classical definition of proba bility. N denotes the total number o f combinations, N = 󰇡   󰇢 , when p numbers are randomly drawn from a total of n numbers. M t denotes the number of all possible combinations with a t-hit. It is given by        󰇡   󰇢 . (1) The total number N of combinations can alternatively be obtained with the sum over M t :  M t = N p t=0 . (2) 4/16 Table 1 lists all possible combinations M t with a t -hit with their corresponding probabilities P (t) for the 6/49 lot tery, with n = 49, p = 6, and index t = 0, …, 6. The probabilities match with the experience of players. In the majority of cases, 0 -hits up to 2-hits are achieved. Sometim es 3-hits occur, and 4-hits are rare. A lot of luck is required to achieve a 5-hit, a nd it is more likely to be struc k by lightni ng than to a chieve a 6 -hit and win the jackpot. Table 1: All combinations M t with a t-hit and probabilities P(t) for the 6/49 lottery. In the following se ctions, we elaborate on probabilities to achieve 5-hits and the 6-hit with many tic kets. I f required, the theor y can be e xtended including 4-hits or even 3 -hits. By changing the parameters n and p, the theory can be adapted to other lotteries. 2.1.1. The probabilistic method with unique tickets avoiding doubles Let us now discuss the strategy in whi ch w e pick v ti ckets at random, but making sure that each tickets is unique (no doubles, all tickets have to be pairwise different). M t denotes the number of all possible combinations with a t -hit, a s provided in equation 1. I t is obvious that by filli ng out more than just one ticket, one might a chieve more than just one t-hit listed in table 1. A more generalized theory needs to be applied, tailored to the quantity v of tickets. P(m t , v) defines the probability to achieve exactly m t t-hits, when v tickets are choosen at random with no doubles. It is given by the hypergeometric distribution t 0 1 2 3 4 5 6 Mt 6,096,454 5,775,588 1,851,150 246,820 13,545 258 1 P(t ) 43.596% 41.302% 13.238% 1.765% 0.097% 0.002% 0.0000072 % 5/16  󰇛     󰇜  󰇡     󰇢󰇡     󰇢     . (3) In order to explore the structure in equation 3, we may check familiar cases. F or inst ance, P(m t = 1, v = 1) is the probability to achieve exactly one t -hit with a single ticket. I t calculates to  󰇛        󰇜  󰇡    󰇢󰇡    󰇢          . (4) The fraction M t /N on th e right hand sid e in equ ation 4 is the prob ability P(t) already introduced in the previous section. If we want to know the probability to achieve the 6 - hit and several 5 -hits with a c ertain quantity v of tickets, the multivariate hypergeometric distribution is required. P(m 5 , m 6 , v) denotes the probability to achieve m 5 5-hits and m 6 6-hits with v tickets:  󰇛        󰇜  󰇡     󰇢󰇡     󰇢󰇡         󰇢     . (5) M 5 = 258, m 5 can take values in the range 0 … 258 , depending on how many tickets v are filled out. S ince M 6 = 1, m 6 c an only take the values 0 or 1. Equa ti on 5 serves to calculate the complementary prob ability for achieving a minimum number of t -hits. At first, we ask for the probability that we achieve no 5-hit and no 6-hit at all with v tickets. With equation 5 an d m 5 = 0, m 6 = 0, we get  󰇛          󰇜  󰇡    󰇢󰇡    󰇢󰇡      󰇢                 . (6) The fraction on the right hand side of eq uation 6 does consist of two binomi al coefficients, which for larger values of v are challenging to compute. There are two useful approximations of equation 6, for small v and for small (M 5 + M 6 ). Condition v < N – (M 5 + M 6 ): 6/16  󰇛          󰇜              󰇡       󰇢  . (7) Condition (M 5 + M 6 ) < ( N – v):  󰇛          󰇜              󰇡    󰇢     . (8) The approximation in equation 7 is identical to the expression gained for filling out tickets at random, allowing for doubles (see equation 11 in the following chapter). As long as the condition v < N – (M 5 + M 6 ) holds, the number of tickets is sufficiently small that it doesn´t make a difference if tickets are unique without doubles or not. For the conditions of the 6/49 lottery and lar ger qua ntities v of tickets, equation 8 is better suited, since (M 5 + M 6 ) = 259, which is small co mpared to ( N – v). The complementary probability to achieve at least one 5-hit with v tickets is given by  󰇛      󰇜     󰇛          󰇜    󰇡    󰇢     . (9) To be more spe cific about the argument in brackets provided in P(m 5 ≥ 1, v): a chieving at least one 5- hit includes 1 or 2 or 3 … or finally all M 5 = 258 combinations with 5-hits or the 6-hit (with M 6 = 1 there is only one 6-hit). One should internalize th e difference between “exactly” one 5- hit, and “at least” one 5- hit. The proba bilit y to achieve exactly one 5-hit with v = 60,000 tickets is calculated with equation 3, which yields P(m 5 = 1, v = 60,000 ) ≈ 37 %. In contrast to that, the probability to achieve at l east one 5-hit with 60,000 tickets is calculated with equation 9, which yields P(m 5 ≥ 1, v = 6 0,000 ) ≈ 67%. The value is higher, because it is not li mited to just one 5 - hit. It does include other possible events such as two 5-hits and no 6-hit up to 258 5-hits and the 6-hit. Equation 9 is applied to determine the number of ti ckets required to achieve specific probabilities for at least one 5-hit. Some values are provided in table 2. 7/16 Table 2: Probabilities to achieve at least one 5-hit related with the number of tickets. The expression for the 6-hit is derived from equation 9. With M 5 = 0 and M 6 = 1 we get  󰇛      󰇜     󰇛      󰇜    󰇡    󰇢     . (10) With unique ti ckets (no doubles, a ll blocks have to be pa i rwise different), the probability to win the jackpot scales linearly with the number of tickets v. The identical re sul t can b e obtained with equation 3. The only chance to secure th e jackpot with no risk at all is “ buying the pot ” , which st ands for submitting all possible combinations, v = N. The methods above can be extended in a straightforward manner including 3-hits and 4- hits. One just needs to extend equations 5 to 9 with M 3 and M 4 . By changing the parameters n and p, the theory can be adapted to other lotteries. 2.1.2. The probabilistic method with tickets allowing for doubles Let us now discuss the strategy in which we pick v ti ckets at random, without making sure that each tickets is unique, allowing for doubles. In other words: the player doesn´t care about the numbers on previously filled out tickets. It simplifies the reasoning to switch the parameters from M t and N to the probabilities P(t). The probability to achieve a 5 -hit or the 6-hit with a single ticket is P (5) + P(6). The complementary probability to a chieve no 5 -hit and no 6-hit with a single tic ket c alculates to 1 – P(5) – P(6), and the probability to achieve no 5-hit and no 6-hit with v tickets is Nu m ber of ti cke ts v 1 10 100 1,000 10,000 100,000 300,000 500,000 Proba bi l i t y f or at l eas t on e 5- h i t P(m 5 ≥ 1, v ) 0.002% 0.019% 0.185% 1.835% 16.913% 84.414% 99.636% 99.992% (49 , 6, 6, 5) l ott er y sch em e 8/16  󰇛          󰇜  󰇟    󰇛  󰇜  󰇛󰇜 󰇠  . (11) Equation 11 is identical to equation 7. The proba bil ity P(m 5 ≥ 1, v) to achieve at least one 5-hit with v tickets calculates to  󰇛      󰇜    󰇟    󰇛  󰇜  󰇛󰇜 󰇠  . (12) For the 6/49 lotter y, the results obtained with equation 12 differ only marginally from that obtained with equation 9 in table 2. For targeting at least one 5 -hit, tickets can be filled out allowing for doubles or not. The expression for achieving the 6-hit is  󰇛      󰇜    󰇟   󰇛󰇜 󰇠  . (13) Assuming that i t was possible to submit 5 mi llion t ickets allowing for doubl es, we obtain the result briefly discussed in the introduction:  󰇛           󰇜    󰇣        󰇤         . However, for increasing the probability to achieve the 6- hit it is smarter to fill out tickets with no doubles. Equation 10 yields  󰇛           󰇜               . 2.2. Tackling the 6/49 lottery with combinatorial designs 2.2.1. The lottery design with benchmark An (n, k, p, t) lot tery sc heme consists of an n- se t of elements V = {1, 2, 3, …, n} , a collection of k-element s ubsets of V (called k-subsets or blocks), and a p-element subset of V (called p -subset). T he parameters n, k, p, t m ust satisfy the relationship t ≤ {p, k} ≤ n. The k-subsets, or blocks, are designated as a lottery design LD(n, k, p, t) when any p- 9/16 subset of V interse cts at lea st one of these blocks in at least t elements. The smallest quantity of blocks of the lottery design LD(n, k, p, t) is denoted as lottery number L(n, k, p, t) [2]. In lotteries, the blocks correspond to the tickets. Large lottery designs to address the 6/49 lottery haven´t been around for long, because they are challenging to co mpute. Before explicit designs came up, the lower and the upper bounds for the lottery number L(49, 6, 6, 5) were deter mi ned to at le ast narrow down the challenge. In this c ontext , the work of W. R. Gründling is significant, because it provides a survey of ac hievements up to 2004 [3 ]. L(49, 6, 6, 5) = 62,151 is provided as the lower bound, which is the number of tickets to be filled out for a chieving at least one 5-hit. This lower bound h as not been reached so far. The current record for the lottery d esign LD(49, 6, 6, 5) was uploaded in 2024 into the online database “ Covering R epository ” [4]. I t does contain 142,361 blocks, which is the numb er of t ickets to be submitted for achieving at least one 5-hit with absolute certa inty. The lottery design should be benchmarke d with equations 9 and 10, where the same number of ti ckets was filled out at random, avoiding doubles . With 142,361 ticket s the probability to achieve at least one 5 hit calculates to  󰇛          󰇜    󰇡          󰇢       . With 142,361 ticke ts , the residual risk of ~7% to miss the 5-hit might look acceptable for bold players, if they woul d dispense with the lottery design, and rather try the probabilistic method. For the 6-hit, we obtain from equation 10:  󰇛          󰇜             . When t he lower limit of 62,151 blocks mi ght be reached for the LD(49, 6 , 6, 5) lottery design at some point in t he future, things look different. The probability to achieve the 6- 10/16 hit will decrease, for the lottery design as well as for the probabilistic method, because it just scales with the number v of submitted ti ckets. The proba bilit y to achieve at least one 5-hit would be  󰇛          󰇜    󰇡          󰇢       . In this case, applying the lottery design was a cl ear adv antage, if the player was able to submit 62,151 tickets, and if the profit was high er than th e costs for submi ssion. The probabilistic method just guarantees a probability of ~68%, whereas the lottery design guarantees an absolute certainty to achieve at least one 5-hit. 2.2.2. The covering design with benchmark A (n, k, t) covering design is a collection of k-element subsets (called k -subsets or blocks) of an n- set of e lements V = {1, 2, 3, …, n} su ch that any t- element subset (call ed t-subset) is contained in at least one block. The parameters n, k, t must satisfy the relationship t ≤ k ≤ n. The smallest quantity of blocks of the (n, k, t) covering design is denoted as covering number C(n, k, t) [5, 6]. In lotteries, the blocks correspond to the tickets. Covering designs and lottery designs are related to e ach other [7]. Adding the trivial case that all k-subsets of an n-set, with gives N =     combinations in total, do provide the biggest covering design, L(n, k, p, t) and C(n, k, t) fulfill the inequalities 󰇛      󰇜  󰇛  󰇜      . (14) By construction covering designs are larger than lottery designs. I n contrast to lot tery designs, which just guarantee at least one 5-hit, co vering designs are complete insofar as they guarantee at least all six 5-hits of a random draw of 6 numbers in th e 6/49 lottery . 11/16 Covering designs can be obtained from “The La Jolla Covering Repository ” , a comprehensive online d atabase fo r combinatorial structures [8]. The (49, 6, 5) covering design was constructed a nd added to the database in February 2026 [9 ]. It is built on the groundwork laid by R. H. F . Denniston and A. Jurcovich [10, 11 ]. C(49, 6, 5) = 325,205 does meet the Schönheim bound, which denotes th e lowe r limit of covering designs [12]. That means: i n order to cover all six 5-hits in the 6/49 lottery, which includes the case that a t least one 5-hit is a chieved, one has to submit the 325,205 tickets as provided in the (49, 6, 5) covering design. There is no smaller, no more efficient covering design. As accomplished in the previous chapter with the lottery design, the covering design is benchmarked with a random test with equations 9 and 10 . If 325,205 tickets were filled out at random avoiding doubles, the probability to achieve at least one 5 hit calculates to  󰇛          󰇜    󰇡          󰇢       . The probability for the 6 hit just scales with the number of ti ckets submitted , for the probabilistic method and for the covering design alike. From equation 10 we get:  󰇛          󰇜             . With 325,205 tickets, the residual risk of ~0.2% to miss any 5-hit is negligible – if we dispense with the covering design and go for the probabilistic method. However, the covering d esign does guarantee at least all six 5- hits in the 6/49 lottery. When choosing random tickets avoiding doubles, such an event does happen less often. To cover at least all six 5 -hits calculates to P(m 5 ≥ 6, v = 325,205) ≈ 56% , which is lower. If the prize for 5-hits was high enough and bulk submission wa s still possible, one should s ubmi t tickets filled out according to the (49, 6, 5) covering design. 12/16 Lottery agencies have prohibited bulk submissions, and the prizes for 5 -hits ensure that the expe cted pro fit is not worth the efforts. Alth ough the (49, 6, 5 ) covering design is finally available, it s relevance is purely academic – due to its efficiency a mathematical beauty, so to speak. It might be applied as a building block for other designs of interest. 2.3. Debunking myths On occasion, jackpot w inners do attribute their luck to birthday numbers or to any number patterns on tickets. People who have never won anything at the lottery do that, too. The mathematics in the previous chapters allows to debunk such myths. Any restriction of numbers to a few lucky numbe rs, to even or to odd numbers, to number patterns such as diagon als across the 7 x 7 square field of 49 numbers, or to the left, to the right, to the upper or to the lower half of the square field can mathematically be t reated as a n*-subset out of 49 numbers. Wh atever its position in the squar e field, n o number stands out from all the other numbers. In order to qu antify the restriction to n* numbers, we ask: what is the prob ability, that at least 5 of the 6 winning numbers, fall among the n* picks chosen from n = 49 numbers? The logic behind is equal to asking for the probability that a draw of p = 6 balls from an urn of n = 49 balls, which contains n* white balls and (n - n*) black balls, does include t = 5 or t = 6 white balls. The answer is given by the hypergeometric distribution  󰇛    󰇜  󰇡    󰇢󰇡    󰇢 󰇡   󰇢 . (1 5) Equation 1 5 has the same structure li ke equation 3, but one should internalize that the perspective has turned to the p = 6 winning numbers. With t = 6, p = 6, and n = 49 equation 15 yields 13/16  󰇛      󰇜  󰇡    󰇢󰇡     󰇢      󰇡    󰇢      . (1 6) With t = 5, p = 6, and n = 49 equation 15 yields  󰇛      󰇜  󰇡    󰇢󰇡     󰇢      󰇡    󰇢 󰇛    󰇜      . (1 7) That at least one 5 -hit, which includes the 5-hit or the 6-hit, falls among th e n* picks is the sum of equations 16 and 17:  󰇛      󰇜  󰇡    󰇢      󰇡    󰇢 󰇛    󰇜      . (18) If one applies n* = 10 favorite numbers, for instance mixed from various birthdays, the probability that at least one 5-hit falls among the 10-subset calculates to  󰇛        󰇜                󰇛    󰇜        . The probability is e xtrem ely low. It takes effect only if all combinations of the 10-subset, which calculates to     = 210 tickets, are filled out. However, the tickets would be better spent on all 49 numbers! Equation 9 yields  󰇛        󰇜    󰇡        󰇢      . If one focusses onto com binations in one half of the square field of 49 numbers, w hich amounts to n* = 25, the probability that at least one 5 -hit falls among the 25-subset calculates to  󰇛        󰇜                󰇛    󰇜         . One could naively assum e P(t ≥ 5, n* = 25) ≈ 50%, that a fair distribution of luck would scale with the quantity of numbers covered. As so often, when faced with combinatorics 14/16 and probabilities, int uition is wrong. For P(t ≥ 5,n* = 25) = 10.385% to take effect, all combinations of the 25-subset have to be filled out, which calculates to     = 177,100 tickets. Again, the tickets would be better spent on all 49 numbers! Equation 9 yields  󰇛          󰇜    󰇡          󰇢       . If we would apply a (10, 6, 5) covering de sign with C(10, 6, 5) = 50 for the first example, and a (25, 6, 5) covering design with C(25, 6, 5) = 9,321 for the second example, calculations turn out that it was still better to follow the probabilistic method and spend the tickets on all 49 numbers. The key lesson: limiting lottery numbers to prefer red subsets disproportion ately redu ce s the probabilities to achieve at least on e 5 -hit. It can even annihilate the positive effect of covering designs. Focusing onto specific subsets of numbers, or onto any number patterns, just reflects human errors, for which behavioral economists on ce coined the expressions “control illusion” and „simulation error“ . P eople boost their self -efficacy believing they would control the outcome of random events, and a rranging numbers gives the halo of expertise whe re none exists. 3. Conclusion This work combines probabilistic methods to tackle the 6/49 lottery with lottery designs and with covering designs. As expected, none of these methods works miracles. The probabilistic method can be applied as a benchmark for combinatorial designs. If lottery agencies hadn´t closed the loopholes for bulk submissions, and prizes for 5-hits were high enough to justify the efforts, the LD(49, 6, 6, 5) lottery design shoul d further be improved, because L(49, 6, 6, 5) = 62,151 h as not been reached yet. 15/16 The newly constructed (49, 6, 5) covering design is introduced, which wa s not publi cly available before. However, C(49, 6, 5) = 325,20 5 blocks could as well b e filled out at random for achieving at least one 5 -hit with negligibl e risk. Nevertheless, the covering design guarantees at leas t all six 5-hits of a draw of 6 numbers. Even i f t he prize for a single 5-hit was low, the winnings would be paid out at least six times. As mentioned earlier, bulk submissions are not possible any mor e, which makes th e cov ering design just a mathematical curiosity. Due to it s size, i ts relevance is purely scientific, because it meets the Schönheim bound. The only way to win the jackpot with no risk at all is “buying the pot”. The probability for the 6-hit scales with v/N, for probabilistic me thod s and for combinatorial designs alike. Any other promises belong in the realm of phantasy. Common myths referring to lucky numbers or to any other preferred subset s of numbers are debunked. Such met hods can be attributed to the illusion to be able to improve the outcome of random events. Any picking of numbe rs, be it birthday numbers, be it number patterns or input from astrolog y, can mathematically be treated as a preferred n* -subset of the total of n numbers. Calculations demonstrate that limiting numbers to preferences disproportionately reduces the chances of winning. The r estriction c an even a nnihilate the positive effect of covering designs. Disclosure I have no conflicts of interest to disclose. References [1] Crockett Z. The man w h o won the lottery 14 times. The Hustle; 2023. Available from: https://thehustle.co/the-man-who-won-the-lottery-14-times 16/16 [2] Colbourn CJ . CRC Handbook of Combinatorial Designs. Boca Raton: CRC Press ; 1996. Chapter V, Applications, 8 Winning the Lottery; p. 578 – 584. [3] Gründlingh WR . Two New Combinatorial Problems involving Dominating Sets for Lottery Schemes. Dissertation for the degree Doctor of Philosophy at the Department of Applied Mathematics of the University of Stellenbosch, South Africa; 2004. 187 p. [4] The Covering Repository. Available from: http://www.coveringrepository.com/ [5] Gordon DM , Kuperberg G, Patashnik O. New constructions for covering designs. Journal of Combinatorial Designs. 1995; 3 : 269 – 284. [6] Stinson DR . CRC Handbook of Combinatorial Designs. 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