Coins, dice, cards and bags of marbles are how this chapter builds intuition for chance. Toss, roll and draw as many times as you like — each tab compares what theory predicts against what actually happens.
P(E) = favourable outcomes ÷ all possible outcomes — but only when every outcome is equally likely. Toss a coin or roll a die many times and watch the empirical (observed) frequency settle toward the theoretical probability.
Edit how many balls of each colour sit in the bag, or pick an event on a full 52-card deck, and watch the favourable-outcome count and probability update live.
| Colour | Count | P(colour) |
|---|---|---|
| Red | ||
| Blue | ||
| Yellow |
A blue die and a grey die give 36 equally likely ordered pairs. Choose a condition on the sum and the matching cells light up — count them to get P(E).
Every event E has an opposite, "not E", written E̅. Together they always cover every outcome, so P(E) + P(E̅) = 1 — and no probability can ever sit outside 0 and 1.
When outcomes form a continuum — a length of time, a patch of land — we compare measures (length or area) instead of counting outcomes.
Music stops at a random instant between 0 and 2 minutes. Drag the marker to set the "stop within this long" question and see the probability as a fraction of the line.
Adjust the lake's width and height inside the search rectangle; the probability the helicopter crashed in the lake is lake area ÷ total area.