Lottery Odds (6/60)

Mega-Sena odds (6 of 60)

Brazil's Mega-Sena asks you to pick 6 numbers out of 60. Count every way that can be done and you get C(60,6) = 50,063,860 combinations, which puts a single ticket at about 1 in 50 million for the jackpot. The smaller prizes are easier. Match 5 (Quina) and the odds are C(6,5)·C(54,1) / C(60,6) ≈ 1 in 154,518; match 4 (Quadra) and it's around 1 in 2,332. CAIXA pays back roughly 35% of what it collects, so on average each ticket is worth less than you paid. Play long enough and you lose money. The one exception is a big rollover, when accumulated prizes nudge expected value toward zero, though it almost never crosses into positive territory.

Applications

It shows up in financial education and gambling literacy, in teaching applied combinatorics and probability, and in pointing out cognitive biases like the lottery bias, the availability heuristic, and the "near-miss" effect. People also use it to weigh the expected value of lotteries against other investments, and reporters lean on it to explain how a massive jackpot still leaves each ticket with vanishingly small odds.

FAQ

Does picking 7 or more numbers improve my odds? Yes, in proportion to how many extra combinations you're buying. The catch is that the price climbs just as fast. Ten numbers cost 210× more and buy you a 210× better chance, so you gain nothing per real spent.

Are some numbers "luckier"? No. Every combination is exactly as likely as any other, and what came up last week tells you nothing about the next draw.

Can a lottery ever have positive expected value? On paper it can, during a huge accumulated jackpot. In practice, the chance of several winners splitting the pot plus the tax bite usually drags the real expected value back below zero.