Naive Decision Making : Mathematics Applied to the Social by T W Körner

By T W Körner

Takes readers on an stress-free trip via many elements of mathematical selection making; the one prerequisite is trouble-free calculus.

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A (in other words, Pr(A) is the sum of all the p(ω) with ω ∈ A). We call Pr(A) the probability of the event A. In our horse race example, if the first k horses are brown and the rest black, then taking the event A to be the set of points corresponding to a brown horse winning we have Pr(A) = p(ω1 ) + p(ω2 ) + · · · + p(ωk ). It is remarkable that such a simple set of rules gives rise to such a rich theory. 1 We use the notation above. (i) Show that, if ω ∈ , then Pr({ω}) = p(ω). (ii) Show that, if A and B are events and A ⊇ B, then Pr(A) ≥ Pr(B).

We give p j , t j and T their usual meanings. We assume that p1 /t1 ≥ p2 /t2 . (i) Show that if we bet y on the first horse and nothing on the second, our expected profit is f (y) = T − p1 t1 (T + y) + p2 (T + y) . t1 + y (ii) Show that f (y) = p1 t2 − p2 y − p 1 t1 t 2 . t1 + y By considering the first and second derivatives, show that f (y) increases as y increases from 0 to Y0 where p 1 t1 p2 = , 2 t2 (t1 + Y0 ) but that the rate of increase f (y) of f decreases as y increases. (iii) Show that, when y > Y0 , f (y) decreases as y increases.

It is the separate job of statisticians to attempt to apply ‘probability theory’ to the real world and that of philosophers to worry why such an application can be made. In this section we set out the rules for a simple theory of probability. 3 If we consider a horse race with n horses, then we might take = {ω1 , ω2 , . . ωn } with ω j the point corresponding to the jth horse winning. We also have a function p : → R such that p(ω) ≥ 0 for each ω ∈ and p(ω) = 1. ) In our horse-racing example, p(ω j ) would be the probability that the jth horse wins the race.

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