Between Impossible and Certain The Mathematics of Maybe: Introduction to Probability

Two friends can genuinely disagree about whether it will rain today, each with their own reasonable-sounding prediction -- this everyday uncertainty is exactly what probability turns into something measurable. Every event, whatever the source of its uncertainty, can be placed somewhere on a single scale running from 0 (completely impossible) to 1 (completely certain), with 0.5 marking an even, fifty-fifty chance -- rolling higher than 6 on an ordinary die sits at 0 (impossible), while drawing any card at all from a full deck sits at 1 (certain, since every card is guaranteed to come from the deck).

Experimental probability is built from real, actually-run trials: rolling a die 50 times and getting a 4 exactly 8 times gives an experimental probability of 8 divided by 50, or 0.16 (16 percent) -- also called the relative frequency. Theoretical probability instead reasons about the situation directly, without running any trials at all: since a die has 6 equally likely faces, and only one of them is a 4, the theoretical probability of rolling a 4 is exactly 1 divided by 6, about 0.167 (16.7 percent) -- genuinely close to, but not identical to, the experimental result, since a real experiment's outcome always carries some natural variation around the theoretical value.

Rolling three consecutive 6s in a game of Snakes and Ladders might feel like it makes a fourth 6 less likely, purely out of a sense that 'it's due to change' -- but this feeling, known as the gambler's fallacy, is genuinely mistaken: a fair die has no memory of its previous rolls, so the probability of rolling a 6 stays fixed at exactly 1 divided by 6 (about 16.6 percent) every single time, completely independent of whatever happened on the rolls before it.

The sample space is the complete list of every possible outcome an experiment could produce -- tossing two coins together gives the sample space {HH, HT, TH, TT}, four equally likely outcomes in total. An event is simply any subset of that sample space that you're interested in -- 'at least one head' from those two coin tosses picks out exactly three of the four outcomes (HH, HT, TH), giving that event a probability of 3 divided by 4.

A tree diagram lays out every stage of a multi-step experiment as a set of branches, each one labelled with its own probability -- tossing a coin twice draws two branches from the start (H or T, each probability one-half), and each of THOSE branches splits again into two more (H or T, again one-half each), giving four final branch-ends matching the sample space {HH, HT, TH, TT}, each reached by multiplying along its own path: one-half times one-half equals one-quarter for each specific outcome.