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The new probabilities are just those of the arrival times of the 4 and the 6. For example, assume that all the original trams arrive in equal proportions. Then in the original collective the relative frequency of 4’s is 1/5. But in the new collective it is 1/2. 1). 16 probability and relative frequencies The fourth operation concerns the combination of collectives (and so is rather unsurprisingly known as combination). The standard example concerns rolls of dice, but it’s very boring. Instead, let’s join Prokop, a fan of Bogart, in the back room of Rick’s Café Américain.

We might not know the actual frequency, but that’s a matter for statistical inference (for example of the kind discussed in Chapter 3). This probability obeys the axiom, and presumably could be useful in certain circumstances—other examples might be voting preferences or intentions to buy particular consumer products. Still, such finite frequencies would not be about mass phenomena, and so would not be a substitute for the limiting relative frequency interpretations or the propensity interpretations covered in the next chapter.

For example, a roll of the die can come up either 1 or 6, but not both. Suppose that the die is fair, that is, after repeated rolls, n(1)/n = n(6)/n = 1/6. Then the probability that either 1 or 6 will come up should be higher, there being more events to be counted in the numerator. The number of times the die comes up 1 or 6, n(1 or 6), is, of course, the number of times the die comes up 1 plus the number of times the die comes up 6, n(1) + n(6). This reasoning leads us to the third axiom that the prob­ abilities of exclusive events add: (3) p(A ∪ B) = p(A) + p(B), if A ∩ B = ∅.

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