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The set \(S\) is the set of all integers from 1 to \(n\). The set \(T\) is the set of all distinct subsets of \(S\), including the empty set \(\emptyset\) and \(S\) itself. Show that \(T\) contains exactly \(2^n\) sets. The sets \(A_1, A_2, \ldots, A_m\), which are not necessarily distinct, are chosen randomly and independently from \(T\), and for each \(k\) \((1 \leq k \leq m)\), the set \(A_k\) is equally likely to be any of the sets in \(T\).

1. Write down the value of \(P(1 \in A_1)\).
2. By considering each integer separately, show that \(P(A_1 \cap A_2 = \emptyset) = \left(\frac{3}{4}\right)^n\). Find \(P(A_1 \cap A_2 \cap A_3 = \emptyset)\) and \(P(A_1 \cap A_2 \cap \cdots \cap A_m = \emptyset)\).
3. Find \(P(A_1 \subseteq A_2)\), \(P(A_1 \subseteq A_2 \subseteq A_3)\) and \(P(A_1 \subseteq A_2 \subseteq \cdots \subseteq A_m)\).

The number \(X\) of casualties arriving at a hospital each day follows a Poisson distribution with mean 8; that is, \[ \P(X=n) = \frac{ \e^{-8}8^n}{n!}\,, \ \ \ \ n=0, \ 1, \ 2, \ \ldots \ . \] Casualties require surgery with probability \(\frac14\). The number of casualties arriving on any given day is independent of the number arriving on any other day and the casualties require surgery independently of one another.

1. What is the probability that, on a day when exactly \(n\) casualties arrive, exactly \(r\) of them require surgery?
2. Prove (algebraically) that the number requiring surgery each day also follows a Poisson distribution, and state its mean.
3. Given that in a particular randomly chosen week a total of 12 casualties require surgery on Monday and Tuesday, what is the probability that 8 casualties require surgery on Monday? You should give your answer as a fraction in its lowest terms.