Calculate probabilities for coin flips, dice rolls, card draws, and custom events with combination support.
Enter a scenario such as a number of coin flips, dice rolls, or card draws, plus the outcome you want, and the tool returns the probability of that outcome. It counts equally likely outcomes and uses combinations, so repeated trials such as 3 heads in 5 flips and hands drawn without replacement such as 2 aces in 5 cards are both handled, along with custom events where you set your own per-trial success probability. The most common surprise is the gap between exactly and at least: exactly 3 heads in 5 flips is 31.25%, while at least 3 heads is 50%.
A concrete input and expected output from the current implementation.
Input
5 coin flips, exactly 3 heads, fair coin
Expected output
Probability: 0.3125 (31.25%)
Each flip has 2 outcomes, so 5 flips give 2^5 = 32 equally likely sequences. The number of sequences with exactly 3 heads is the combination C(5,3) = 10, so the probability is 10/32 = 0.3125.
For a fair coin or die, every full sequence of flips or rolls is equally likely, so the tool divides the number of sequences matching the stated target by the total: 2^n sequences for n flips, 6^n for n rolls. The answer depends entirely on how the target is phrased; exactly 3 heads and at least 3 heads are different counts in the same 5-flip scenario, and the result shown follows the wording you entered.
When order does not matter, the tool counts outcomes with combinations, n choose k = n! / (k! (n-k)!). A 5-card hand from a 52-card deck is one of C(52,5) possible hands, so the probability of exactly 2 aces is C(4,2) x C(48,3) / C(52,5). For a custom event with per-trial success probability p in n trials, the probability of exactly k successes is C(n,k) p^k (1-p)^(n-k).
Because the tool counts sequences, not average heads. Of the 32 equally likely 5-flip sequences, 10 have exactly 3 heads, which gives 31.25%; another 6 have 4 or 5 heads, so at least 3 is 16/32 = 50%. The lower figure answers the narrower question you asked, and the tool shows the larger number when you ask for at least 3.
No. The calculator treats every flip as an independent trial with a fixed 50% chance, and it does not accept past outcomes as input. Five tails in a row is a 1-in-32 event for a fair coin, but the sixth flip still has exactly a 50% chance of heads, because past flips do not influence the next one.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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