Probability basics: events, AND, OR, independence
Before computing anything fancy, let's master the basic language of chance. Just a few simple rules — how to combine probabilities of events: add them, multiply them, take the complement — underpin almost everything that follows, from A/B tests to spam filters. We'll work them out on a clear example: a roll of a die with six equally likely outcomes.
Click A/B under the outcomes to change the events — the diagram and numbers recompute live. On the Venn: the left crescent is only-A, the right only-B, the middle lens is the intersection (A and B), outside are outcomes in neither event.
Take a die roll: six equally likely outcomes. An event is a set of outcomes you care about: "even" = {2,4,6}, "greater than three" = {4,5,6}. Use the A and B buttons under the cells to define two events (by default A is the evens, B is above-three). The Venn diagram above lays out all six outcomes: the left circle is A, the right is B, the shared lens holds outcomes that fall into both. The probability P(A) is the share of the event's cells: 3 out of 6 = 0.5.
These rules are not school formalities — they are an analyst's daily arithmetic. "The chance that at least one of 1000 users hits a rare bug (p=0.1%)" is computed through the complement: 1 − 0.999¹⁰⁰⁰ ≈ 63%. Intuition fails here; the rule does not.
Multiplying independent probabilities also explains multiple comparisons (20 tests, each with a 5% chance of a false alarm, → the chance of at least one false alarm is 1 − 0.95²⁰ ≈ 64%) and why long coincidences are suspicious. Independence is the key — and often false — assumption, always worth a look.
System reliability: failure probabilities of duplicated independent nodes multiply, which is why redundancy slashes risk — fault-tolerance engineering stands on this.
Games and lotteries: jackpot odds are a product of independent probabilities, hence their astronomical rarity. The same rules power risk pricing in insurance.