You flipped 6 heads in a row. The coin has no memory — flip 7 is still 50/50. Independent events don't care what came before, and forgetting that has bankrupted gamblers for centuries.
Two events are independent if one happening doesn't change the probability of the other. For independent events, P(A and B) = P(A) × P(B).
Reliability (will two backups both fail?), genetics, repeated trials, security (independent factors), and dodging the gambler's fallacy.
Flip a coin and watch heads/tails settle towards 50/50.
Goal: 50% heads, exactly. The more flips, the closer you get.
This multiplication rule holds ONLY when A and B are independent. If knowing B changes P(A), they're dependent — use conditional probability instead.
Roll two dice. P(both show 6)?
A system has two independent backups, each 99% reliable. Chance both fail?
0.01 × 0.01 = 0.0001 = 1 in 10,000. Independence is why redundancy works so well.
Drawing cards without replacement is NOT independent — the first draw changes what's left. Independence requires the events to genuinely not influence each other.
The gambler's fallacy ('red is due after 5 blacks') and the hot-hand fallacy ('he's on a streak, bet on him') are opposite errors — both born of refusing to believe independent events have no memory.
- Independent = one event doesn't affect the other's probability.
- Then P(A and B) = P(A) × P(B).
- 'Without replacement' breaks independence; past results never matter for truly independent trials.