Every Outcome Gets an Equal Vote Probability

A coin toss or a die roll is called fair (or unbiased) precisely because every one of its possible outcomes is equally likely -- but not every random experiment has this property automatically: a bag holding 4 red balls and just 1 blue ball does NOT give red and blue an equal chance of being drawn, even though drawing SOME ball at all is still a fair, equally-likely-among-balls kind of random draw. Recognising exactly which outcomes are genuinely equally likely, and which aren't, is the first and most important step before any probability can be calculated.

Theoretical (or classical) probability, credited to the French mathematician Pierre Simon Laplace in 1795, is simply favourable outcomes divided by total equally-likely outcomes -- rolling higher than 4 on a die gives 2 favourable outcomes (5, 6) out of 6 total, a probability of 1/3. This single formula automatically respects two extreme cases: an impossible event (rolling an 8 on an ordinary die) always gives probability exactly 0, and a certain event (rolling less than 7) always gives probability exactly 1 -- meaning every genuine probability must sit somewhere between 0 and 1, with no exceptions.

Drawing an ace from a standard 52-card deck has probability 4/52 (1/13, since there are 4 aces) -- and the probability of NOT drawing an ace can either be counted directly (48 non-ace cards out of 52) or derived algebraically from the first answer, since an event and its complement (everything that ISN'T that event) must always add up to exactly 1 between them. This complementary relationship, P(E) + P(not E) = 1, turns any probability question about 'how often does this NOT happen' into a simple subtraction from 1, rather than a fresh count.

Beyond dice and cards, probability can measure a genuinely continuous, geometric situation too: music stopping at a random moment within the first half of a 2-minute window gives a probability of exactly 1/4 (half a minute out of the full 2), found not by counting discrete outcomes but by comparing lengths directly. A similarly-flavoured problem, locating a missing search party somewhere within a rectangular region containing a smaller lake, compares AREAS instead of lengths -- the lake's own 3km by 2.5km footprint against the full 9km by 4.5km search region, giving a probability of exactly 5/27 that the party is specifically within the lake.

Throwing two dice together gives 36 equally likely outcome-PAIRS (6 faces on each die, independently), but the SUMS those pairs produce are genuinely not equally likely at all -- a sum of 7 can be reached six different ways (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), while a sum of 2 or 12 can each be reached only one single way, giving 7 a probability of 6/36 while 2 and 12 each get only 1/36. This is a genuinely important distinction: equally-likely BASE outcomes (the individual dice-pairs) do not automatically make every DERIVED outcome (the sums) equally likely too.