Chapter 7

Measuring Uncertainty

The study of the rules of probability is interrupted in order to deal with an important, outstanding issue: the measurement of uncertainty. The method of comparison with a standard, which was used to obtain the rules, is rarely satisfactory and other methods need to be developed.

# 7.1 Classical Form

With any event E is associated the complementary event E ^{c} that is true whenever E is false, and false whenever E is true. It was shown in §3.7 that your two probabilities for these events necessarily add to one: . It follows that the measurement of the uncertainty of any event may be replaced by that of its complement because one probability can be calculated from that of the other. We saw in §5.5, an example involving birth dates where this was advantageous. Here we study the special case where your beliefs in the event and its complement are the same; p (E) = p (E ^{c}). In that case, since they add to one, both probabilities must equal one half; p (E) = p (E ^{c}) = ½. An example is provided by the genuine toss of what appears to you to be a coin from a reputable mint, where your belief that it will land heads equals that for tails; hence both events, “heads” and “tails”, have probability one half. Notice that there is no obligation on you to have the same beliefs in the two outcomes, only that if you do, your probabilities are both one half. The idea extends ...