Exercise 1.46

Question

Let $A_{1}, A_{2}, \ldots, A_{n}$ be events. Let $B_{k}$ be the event exactly $k$ of the $A_{i}$ occur, and $C_{k}$ be the event that at least $k$ of the $A_{i}$ occur, for $0 \leq k \leq n$. Find a simple expression for $P\left(B_{k}\right)$ in terms of $P\left(C_{k}\right)$ and $P\left(C_{k+1}\right)$.

fifthist x · Accepted Answer

$B_{k} = C_{k} - C_{k+1}$. Since $C_{k+1} \subseteq C_{k}$, 
$P(B_{k}) = P(C_{k}) - P(C_{k+1})$.

Exercise 1.46

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Exercise 1.46

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