Independent Events
Introduced| English |
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| independent events/ˌɪndɪˈpendənt ɪˈvents/ |
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:
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$$P(A \cap B) = P(A)\,P(B)$$
- Since $P(A \mid B) = P(A)$, the condition drops out — just multiply.
- This "and = multiply" only holds when events are independent.
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.
Independent and mutually exclusive are not the same — they're opposites. Disjoint events can't co-occur, so learning one happened tells you the other didn't — that's maximal dependence. Only use $P(A\cap B)=P(A)P(B)$ after you've checked independence; never assume it just because the problem gives you $P(A)$ and $P(B)$.
Flip a fair coin twice (independent flips).
- $P(\text{two heads}) = P(H)\,P(H) = 0.5 \times 0.5 = 0.25$.
- $P(\text{at least one head}) = 1 - P(\text{no heads}) = 1 - 0.5 \times 0.5 = 0.75$.
- The complement turns a messy "or" into a clean "and."
Events are independent when $P(A \mid B) = P(A)$; then $P(A \cap B) = P(A)\,P(B)$ (multiply). This is not the same as mutually exclusive (which means $P(A\cap B)=0$) — they're opposites. Combine the addition and multiplication rules for compound events, often using the complement for "at least one."
Independent events branch by branch
For independent events, multiply along the branches.
Two independent events have P(A)=0.5 and P(B)=0.5. Find P(A and B).
Independent → multiply: 0.5 × 0.5 = 0.25.
Independent events and mutually exclusive events are the same thing.
They're opposites: disjoint events are actually dependent.
Flip a fair coin twice. Find P(at least one head).
1 − P(no heads) = 1 − 0.5×0.5 = 0.75.
Events A and B are independent means...
Independence: the condition doesn't change A's probability.
For 'at least one', it is often easiest to use the ___ rule (1 minus P(none)).
P(at least one) = 1 − P(none).