10.4 Independent events
Definition 10.25. Two events \(A\) and \(B\) from the same sample space are independent if \[P(A\cap B)=P(A)P(B).\]
Note 10.26. Independence means that neither event affects the other, so \(P(B\mid A)=P(B)\) — being told \(A\) happened changes nothing. Substituting that into the “and” rule of the previous section gives exactly the definition above. The mango crate is the contrast: there the first pick did change the second, so the two events were not independent and the plain product \(\frac {3}{8}\times \frac {3}{8}\) would have been wrong.
Example 10.27. (a) The probability that person \(A\) will be alive in \(20\) years is \(0.6\), and that person \(B\) will be alive in \(20\) years is \(0.45\). Find the probability that both will be alive in \(20\) years.
(b) A fair die is thrown twice. Find the probability of getting a \(5\) or a \(6\) on the first throw and a multiple of \(3\) on the second.
Solution. (a) The two lives are unconnected, so the events are independent: \[P(A\cap B)=P(A)P(B)=0.6\times 0.45=0.27.\]
(b) A die has no memory, so the throws are independent. On the first throw two of the six faces are a \(5\) or a \(6\). On the second, the multiples of \(3\) are \(3\) and \(6\), again two faces: \[P=\frac {2}{6}\times \frac {2}{6}=\frac {4}{36}=\frac {1}{9}.\]
Note 10.28. Note the contrast with the addition rule. “And” with independent events multiplies; “or” adds. Since probabilities are at most \(1\), multiplying makes the answer smaller and adding makes it larger — a quick check that the right rule was used.
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