Paper 2, Section II, F

Probability | Part IA, 2007

Let A1,A2,,An(n2)A_{1}, A_{2}, \ldots, A_{n}(n \geqslant 2) be events in a sample space. For each of the following statements, either prove the statement or provide a counterexample.


P(k=2nAkA1)=k=2nP(Akr=1k1Ar), provided P(k=1n1Ak)>0P\left(\bigcap_{k=2}^{n} A_{k} \mid A_{1}\right)=\prod_{k=2}^{n} P\left(A_{k} \mid \bigcap_{r=1}^{k-1} A_{r}\right), \quad \text { provided } P\left(\bigcap_{k=1}^{n-1} A_{k}\right)>0


 If k=1nP(Ak)>n1 then P(k=1nAk)>0\text { If } \sum_{k=1}^{n} P\left(A_{k}\right)>n-1 \text { then } P\left(\bigcap_{k=1}^{n} A_{k}\right)>0


 If i<jP(AiAj)>(n2)1 then P(k=1nAk)>0\text { If } \sum_{i<j} P\left(A_{i} \cap A_{j}\right)>\left(\begin{array}{c} n \\ 2 \end{array}\right)-1 \text { then } P\left(\bigcap_{k=1}^{n} A_{k}\right)>0 \text {. }

(iv) If BB is an event and if, for each k,{B,Ak}k,\left\{B, A_{k}\right\} is a pair of independent events, then {B,k=1nAk}\left\{B, \cup_{k=1}^{n} A_{k}\right\} is also a pair of independent events.

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