Lottery probability: where 1 in 50 million comes from

Where 1 in 50,063,860 comes from

The Mega-Sena draws 6 numbers from a pool of 60. You hit the jackpot if your 6 numbers are exactly the 6 drawn, and here is the point that settles the whole calculation: order does not matter. 04-17-23-38-52-60 is the same winning ticket no matter the sequence in which the balls fall. When order does not count, the right mathematical object is the combination, not the permutation.

Substituting n = 60 and k = 6: C(60,6) = 60! / (6! · 54!). The giant factorials cancel, what remains is (60×59×58×57×56×55) divided by 720, which is 36,045,979,200 / 720 = 50,063,860. That is the number of distinct possible tickets. Since only one of them is drawn, the probability of a single bet hitting the jackpot is 1 in 50,063,860, roughly 0.000002%.

Combination

A selection where order does not matter. Lottery is a combination: {4, 17, 23} = {23, 4, 17}.

Arrangement (permutation)

A selection where order matters, like a digit PIN. It yields far larger counts; not the case here.

The lower tiers come from the same formula by counting "partial hits". For the quina (5 correct out of your 6 numbers): you need 5 of the 6 drawn to be on your ticket and 1 of the remaining 54 to complete the draw, C(6,5)×C(54,1) = 6×54 = 324 favorable combinations, or 1 in 154,518. For the quadra (4 correct): C(6,4)×C(54,2) = 15×1,431 = 21,465, or 1 in 2,332. This is the same reasoning the Lottery Odds Calculator applies automatically for any lottery and number count.

Why playing more numbers costs so much

Marking more numbers on the ticket is not "paying a little more": it is buying, all at once, every simple 6-number game that fits inside your chosen numbers. If you mark n numbers, the ticket equals C(n,6) simple bets. Since July 10, 2025 a Mega-Sena simple bet costs R$6.00 (a 20% increase over the previous R$5.00, see how to compute that kind of change in the percentage guide). The total price is simply C(n,6) × R$6.00, and it explodes because combinatorics is factorial, not linear.

Notice what the table really says. Jumping from 6 to 20 numbers multiplies the cost by 38,760 and multiplies the chance by 38,760, the exact same factor. The probability per real spent is constant: each real always buys the same slice, 1/(50,063,860×6), of chance, whether in one huge bet or in dozens of simple games. There is no volume discount on luck. The only record a 20-number ticket sets is its price: R$232,560. And covering all 50,063,860 combinations, the only way to guarantee the jackpot, would cost about R$300 million, almost always more than the prize itself.

Mega, Quina and Lotofácil: orders of magnitude

Each lottery has its own sample space, the number of distinct possible tickets with the minimum bet, which is exactly the denominator of the top prize’s "1 in X". Comparing those sizes is the most honest way to measure "which is hardest". The Mega-Sena has the largest space; Lotofácil the smallest, because marking 15 of 25 leaves relatively few combinations out.

Lotofácil neatly shows why "easier prize" is not "better bet". Hitting all 15 is 1 in 3,268,760, fifteen times likelier than the Mega jackpot, but the top prize is also far smaller, and most of the prize money dilutes into the 11-to-14-hit tiers that land often and pay little. Quina sits in between (C(80,5) = 24,040,016). In all of them the trade-off holds: more accessible prizes carry smaller values, and the share of revenue that returns is the same fraction fixed in law, whatever the game.

What a wheel really guarantees

A wheel (wheeling system, or desdobramento) is a scheme that takes a set of numbers you choose, say 10, and generates a reduced number of games, fewer than the full C(10,6) = 210 combinations, picked to offer a mathematical guarantee. The key phrase is "conditional guarantee". A typical wheel promises something like: "with these 10 numbers across 20 games, if 5 of my 10 are among the 6 drawn, I guarantee at least a quadra". It is a covering theorem, not a luck charm.

1. Read the whole guaranteeEvery wheel guarantee has the form "IF k of my N numbers come out, THEN I secure tier X". Without the "IF", there is no promise.
2. Check the costA wheel generates several games; multiply by the simple bet. The bet cost calculator closes the math before you print.
3. Test against real drawsRun your numbers through the lottery simulator to see, across many draws, what the guarantee delivers, and what it does not.

Where a wheel is honestly useful: when you have already decided to bet on many numbers and want to secure a smaller prize if a good share of them come out, spending less than full coverage. Where it misleads: when sold as a way to "increase your chance of winning the lottery". The chance of hitting the jackpot stays tied to how many numbers you cover, nothing in the arrangement of games changes that. Build and inspect yours in the Lottery Wheel and read the guarantee as coldly as you would read a contract.

The gambler’s fallacy and "overdue" numbers

Each Mega-Sena draw is an independent event: the balls have no memory. The probability that number 13 comes up in the next draw is the same whether it appeared yesterday or has been "overdue" for 200 draws. Statistically, independence means exactly this: the past result does not change the distribution of the next one at all. Spreadsheets of "hot", "cold" and "overdue" numbers describe the past precisely and predict the future with precisely zero accuracy.

The gambler’s fallacy is believing that independent events "correct themselves". A coin that came up heads five times in a row still has a 50% chance of heads on the sixth, it does not "owe" anyone a tail.

Two cognitive traps feed the illusion. The first is mistaking patterns seen in the rear-view mirror for predictive power, the same error the correlation is not causation guide dismantles across all kinds of data. The second is our poor intuition for extremely rare events: the mind does not tell 1 in 50 million apart from 1 in 5 million, so "I almost got it" feels like progress when it is none. It is worth contrasting with games that build in a luck ceiling, the "pity" systems, the math of gacha pity shows a design where the chance genuinely rises with each try. The lottery has no pity: no bet brings you closer to the next one.

• "Overdue" numbers are not more likely, draws are independent.
• Sequences like 1-2-3-4-5-6 have exactly the same chance as any other combination.
• Avoiding "popular" numbers (dates, patterns) does not change your chance of winning, only your chance of splitting the prize.
• No system, lucky number or history "guarantees" the jackpot; the probability is fixed and known.

Expected value: why the house always wins

Expected value (EV) is the average return of a bet if it were repeated infinitely many times. Sum, over all tiers, the probability of each prize times its value, and subtract the cost. If the result is negative, you lose money over the long run, and in the lottery it is structurally negative, not by bad luck, but by legal design.

You do not need to know each prize to know the sign of the EV. Law 13.756/2018, article 16, fixes how much of the revenue becomes prize money: for the numerical-prediction lotteries (Mega-Sena, Quina, Lotofácil), about 43.79% of the total wagered goes to paying prizes, and that slice already includes the income tax withheld on winnings. The rest goes to taxes, social funds, social security, sports and operating costs. This means that, summing every bet, at most about R$0.44 of each R$1.00 returns as a prize. The average return to the bettor is, by construction, less than half of what came in.