Track 4 · Probability · Lesson 4 of 9
Independence
Decide whether two events are independent, use the product rule only when it applies, and never confuse independent with mutually exclusive.
A risk manager multiplies the failure probabilities of two backup generators to claim that a total blackout is a one-in-a-million event. That multiplication is only valid if the generators fail independently — and if both share a fuel supply, they do not. Independence is a strong assumption that must be checked, not assumed.
Two events are independent when knowing that one has happened does not change the probability of the other. If a fair coin lands heads, the next toss is still heads with probability 1/2. If a customer's first name starts with A, that tells you nothing about whether they will renew a subscription.
In the language of conditional probability: and are independent when .
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It has 8 graded exercises with feedback, hints and worked solutions. After it you will be able to:
- Check independence by comparing P(A ∩ B) with P(A)P(B), or P(A | B) with P(A)
- Multiply probabilities for “and” only when the events are independent
- Use the complement to find P(at least one) for independent events
- Explain why mutually exclusive events with positive probabilities are dependent
- Judge independence from a two-way table of real data without confusing dependence with causation