Independent Events

When knowing one tells you nothing

• Two events are independent 独立事件 if one happening doesn't change the other's probability.
• Formally: $P(A \mid B) = P(A)$ — the condition makes no difference.
• Coin tosses are independent: the first result tells you nothing about the second.
• If knowing $B$ does shift $P(A)$, the events are dependent.

The multiplication shortcut

• For independent events, the multiplication rule simplifies:
• $$P(A \cap B) = P(A)\,P(B)$$
• Since $P(A \mid B) = P(A)$, the condition drops out — just multiply.
• This "and = multiply" only holds when events are independent.

A sample space diagram lists every equally likely outcome

Independent ≠ mutually exclusive

• These two ideas are opposites, not synonyms — a classic mix-up.
• Mutually exclusive: they can't both happen ($P(A \cap B) = 0$).
• Independent: they can both happen, and one doesn't affect the other.
• In fact, two events with nonzero probability that are disjoint are always dependent.

Combining rules

• Compound problems chain the addition and multiplication rules.
• "At least one" is often easiest via the complement: $1 - P(\text{none})$.
• $P(\text{none}) = P(\text{not }A)\,P(\text{not }B)\cdots$ for independent events.
• Break the event into "and"s (multiply) and "or"s (add), step by step.

Write from a joint probability of 0.08

• Events A and B are independent.
• P of A is 0.2 and P of B is 0.4. The lesson values 0.25 and 0.75 are another pair.