Independence and Correlation

A coin lands heads five times in a row. Do you expect tails on the sixth flip? Many people feel tails is “due.” But a coin has no memory: as long as it’s fair, the sixth flip is still 50% heads. That’s independence.

What independence means

A and B are independent if learning that A happened doesn’t change your view of B. In symbols, P(B|A) = P(B). Then the chance of both happening is just the product:

P(A and B) = P(A) × P(B)

Two heads in a row, for example: 0.5 × 0.5 = 25%.

Correlation: knowing one changes the other

Many real-world events aren’t independent. Take rain and late food deliveries. Suppose:

• it rains on about 20% of days;
• deliveries are late 30% of the time on rainy days, but only 10% on dry days.

The overall late rate is 0.2 × 30% + 0.8 × 10% = 14%. The true probability of “rain and a late delivery” is 0.2 × 30% = 6%. If you wrongly treated them as independent and multiplied 20% × 14%, you’d get 2.8% — less than half the real figure.

Learning “it’s raining today” moves your estimate of a late delivery from 14% up to 30%. That’s a positive correlation. When one event makes another less likely, it’s a negative correlation — the hotter it gets, the fewer down jackets are sold.

Why it matters for forecasting

• Avoid the gambler’s fallacy. For truly independent events, past results don’t affect the next one.
• Don’t treat linked events as independent. “Three bank stocks all rise next week” — they tend to move together, so multiplying their individual odds badly underestimates the chance.
• Ask “do these move together?” The same weather, the same policy, or the same person’s mood often drives several outcomes at once.