Concepts: Independent events

3.1 When information does not matter

Conditioning changes probabilities because knowing the condition removes outcomes. Sometimes it changes nothing. If the conditional probability of one event given another is the same as its ordinary probability, then knowing the other event tells us nothing, and the two events are called independent. Putting this into the multiplication theorem gives the working test: the probability of the overlap equals the product of the two probabilities. This single equation is what is checked in problems, and it is equivalent to the conditional probability equalling the ordinary probability whenever the condition has nonzero probability. Any of these equivalent statements may be taken as the starting point, depending on what the question supplies.

3.2 Complements and independence

Independence survives replacing one event by its complement, and also replacing both. The proof is short and worth knowing for a two-mark question: the probability of E together with the complement of F is the probability of E minus the probability of the overlap, and substituting the product rule turns this into the product of the probability of E with the probability of the complement of F. The same argument gives the other two cases.

This fact is behind the most useful shortcut in the chapter. The probability that at least one of several independent events happens equals one minus the probability that none of them happens. For two independent events that is one minus the product of the two failure probabilities. Questions about a problem being solved, at least one bulb being defective, or at least one shot hitting the target usually collapse into one multiplication when written this way.

3.3 Two events, three events, many events

For three events, pairwise independence is not enough to call them mutually independent. Mutual independence requires all the pairwise products and also the three-way product. It is possible to build three events that are pairwise independent yet fail the three-way product, so the extra condition is not decoration. In most board problems the word independent is given for the pair involved, and the task is to use the product rule, the complement rule, or both, rather than to prove independence.

3.4 Independence and disjointness are different

Two events that cannot happen together are disjoint, so the probability of their overlap is zero. If both have positive probability they cannot be independent, because the product of the two positive probabilities is positive while the overlap probability is zero. Independence means the occurrence of one gives no information about the other; disjointness means the occurrence of one rules the other out. Confusing these two ideas is the second most common error in the chapter, after reversing a conditional probability.

3.5 A quick check on a standard example

A fair coin is tossed twice. The event that the first toss is a head and the event that the two tosses match are independent, and a quick product check confirms it. But the event that at least one head appears is not independent of the first toss being a tail, because the first event rules out half the cases that the second leaves open. In an exam, the safe method is always the same: name the two events, compute the three probabilities the product rule needs, and compare the overlap probability with the product. That comparison settles the question in one line. When the question says independent, treat it as permission to multiply; when it does not, the product has to be checked against the overlap probability before any multiplication is used.

3.6 Worked example

A lot contains two independent bulbs, and each bulb is defective with probability 0.05. Find the probability that at least one bulb is defective.

The event "at least one defective" is the complement of "both are good". The two bulbs are independent, so the probability that both are good is 0.95 squared, which is 0.9025. The required probability is therefore 1 - 0.9025 = 0.0975. Multiplying the failure probability directly would give the probability that both fail, a different event; the complement converts one into the other. The same pattern answers a problem-solved question: with individual success probabilities p and q, the probability that the problem is solved is one minus the product of the two failure probabilities, and only independence makes the product legitimate.

3.7 A second quick check

A fair die is rolled and a fair coin is tossed, independently. The probability that the die shows an even number and the coin shows a head is one half times one half, which is one quarter. The probability that at least one of the two events happens is one minus the product of the two failure probabilities, and that product is one half times one half again, so the answer is three quarters. The two calculations use the same two facts, the product rule and the complement rule, in the two directions that exam questions ask for.