Probability, at its cleanest, is counting: favorable outcomes over total outcomes. This calculator sticks to exactly countable scenarios — coins, dice, combinations — so every answer is an exact fraction, not a simulation.
Coins: the binomial pattern
Five flips create 2⁵ = 32 equally likely sequences. "Exactly 3 heads" happens in C(5,3) = 10 of them → 10⁄32 = 31.25%. This binomial structure — C(n,k) ways out of 2ⁿ — underlies polling, quality control, and every yes/no repetition in statistics.
Dice: why 7 rules the table
Two dice make 36 ordered pairs. Totals count their combinations: 7 has six ways (1+6 … 6+1) for 16.7%, while 2 and 12 have one way each (2.78%). The neat formula: ways = 6 − |total − 7|. Craps, Settlers of Catan and backgammon strategy all live on this little pyramid.
Combinations vs permutations: does order matter?
Choosing 3 people from 10 for identical seats: C(10,3) = 120. Assigning them distinct roles — president, secretary, treasurer: P(10,3) = 720, exactly 3! = 6 times more, one for each ordering of the chosen trio. The calculator shows both side by side so the distinction stops being abstract.
Fractions, percents, odds — and the fallacy
The same chance wears three outfits: 1⁄13 probability = 7.7% = odds of 1 : 12. And no run of luck rewrites any of them — five heads in a row leaves the sixth flip at exactly 50%, because independent events have no memory. For distribution-based probability (how unusual is a value?), continue to the z-score calculator; the randomness itself lives in the standard deviation calculator's territory.
Exact BigInt counting; results shown as reduced fractions.