Your bets or tries
Binomial probability
- Maths reviewed by Josh Miller
- Tested by Diana Westfield
A binomial probability calculator answers questions of one shape: I’m doing the same thing n times, each try has the same chance, what’s the chance I get k wins? Ten bets on red, twenty penalty-kick shots, a hundred hands of blackjack where you just track wins. Enter the number of tries, the chance each time (or paste in the odds and it fills the chance for you), then the count you care about. You get exactly, at least and at most, plus the whole distribution so you can see where your result sits.
Binomial probability, with a gambling example
Take ten spins on red on a European wheel. Red comes in 18 times in 37, a 48.65% chance. Ask for at least 5 reds and the CasinoCalcs calculator says 58.9%. Ask for at least 7 and it drops to 15.1%, which is roughly the chance of a friend coming back from the casino telling you red was “running hot” over their ten spins. That’s the useful part of a binomial table: it shows how often ordinary luck produces something that feels like a pattern.
It works the other way round too. You’ve backed 30 horses at around 4/1 (a 20% chance each, ignoring the margin) and only three won. The calculator puts at most 3 winners at 12.3%. Unlucky, not unheard of, and certainly not proof you can’t pick a horse. Nine or more winners comes up 12.9% of the time, which is the same kind of luck in the other direction, so a good month at those odds deserves a little suspicion before you double your stakes.
When the binomial fits, and when it doesn’t
It needs four things: a fixed number of tries, each one a win or not, the same chance every time, and no try affecting another. Spins, dice and coin flips tick every box. Sports bets mostly do, as long as you don’t mind treating each match’s chance as known.
Cards are the classic trap. Draw five cards and ask how many are hearts, and every heart you take out changes the odds for the next card. That’s the hypergeometric calculator‘s job, and for one-off deck questions the card probability calculator is quicker. Accumulators are another: the legs aren’t repeats of the same bet, each has its own chance, so use the acca calculator for those.
The maths, if you want it
The chance of exactly k successes is C(n, k) × pk × (1 − p)n − k. The C(n, k) part counts the ways to choose which k tries win; it’s the same number the combination calculator gives. “At least k” adds the exact chances from k up to n. The calculator works with logarithms, so it stays accurate up to 100,000 tries, which is about how many spins a slot player can rattle through in a long month.
The average number of wins is just n × p, and the spread around it (the standard deviation) is √(n × p × (1 − p)). Over 1,000 even-money bets at 48.65% you expect about 487 wins, give or take 16. The probability calculator handles single events and combinations if you need those instead.