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Question Number 147553 by Ar Brandon last updated on 21/Jul/21

	Answered by ArielVyny last updated on 22/Jul/21

	$1) F o r m u l e d e P o i n c a r r e s e d e m o n t r e p a r$ $r e c c u r e n c e$ $2) s o i t f l^{'} a p p l i c a t i o n q u i a u n e l e t t r e i o n$ $a s s o c i e u n d e s t i n a t a i r e o u f (i) e s t l a l e t t r e$ $d u d e s t i n a t a i r e i . c e t t e a p p l i c a t i o n e t a n t$ $b i j e c t i v e c a r d (Ω) = n! d e c e f a i t l a p r o b a b i l i t e$ $d^{'} o b t e n i r l a b o n n e r e p a r t i t i o n e s t P = \frac{1}{n!}$ $3) p r o b a b a b i l i t e p o u r {q u}^{'} u n e l e t t r e a u m o i n s$ $a r r i v e c h e z s o n d e s t i n a t a i r e .$ $s o i t A_{i} = ‘ ‘ l a l e t t r e i a r r i v e c h e z s o n {d e s t i n a t a i r e}^{″}$ $o n v e u t c a l c u l e r P (A_{1} \cap . . . \cap A_{n})$ $- s i u n e p e r s o n n e r e c o i t s a l e t t r e a l o r s$ $l e s (n - 1) a u t r e s p e s o n n e s v o n t r e c e v o i r l e s$ $(n - 1) l e t t r e s r e s t a n t e s$ $a i n s i o n p o s e k e g a l e a u n o m b r e s d e p e r s o n n e s$ $r e c e v a n t c h a c u n e l e u r l e t t r e$ $o n m o n t r e d o n c q u e P = \frac{(n - k)!}{n!} k ⩾ 1$ $4) p r o b a b i l i t e q u e a u c u n e l e t t r e n^{'} a r r i v e c h e z$ $s o n t d e s t i n a t a i r e .$ $o n c o n s i d e r e P = 1 - P^{'} o u P^{'} e s t l a p r o b a b i l i t e$ $p o u r {q u}^{'} a u m o i n s u n e p e r s o n n e r e c o i v e s a$ $l e t t r e a i n s i P = 1 - \frac{(n - k)!}{n!} (k ⩾ 1)$ $5) p r o b a b i l i t e {q u}^{'} a u c u n e l e t t r e n^{'} a r r i v e a$ $d e s t i n a t i o n .$
	Commented by ArielVyny last updated on 22/Jul/21

	$o n n o t e B l^{'} e v e n e m e n t a u c u n e l e t t r e n^{'} a r r i v e a$ $d e s t i n a t i o n A = B^{c} t e l q u e A = \cup A_{i}$ $d^{'} a p r e s < a f o r m u l e d e P o i n c a r r e v u a l a$ $q u e s t i o n 1 o n a P (A) = Σ {(- 1)}^{k - 1} S_{k}$ $a v e c S_{k} = Σ P (A_{i 1} \cap . . . \cap A_{n})$ $a i n s i S_{k} = C_{n}^{k} \frac{(n - k)!}{n!}$ $P (A) = \underset{k = 1}{\sum^{n}} {(- 1)}^{k - 1} \frac{1}{k!}$ $d^{'} o u P (B) = 1 - P (A)$ $P (B) = Σ {(- 1)}^{k - 1} \frac{1}{k!}$
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