- #1

sabbagh80

- 38

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Hi friends,

The problem:

Assume the events [itex] A_i [/itex] for [itex] i=1,2,...,n [/itex] are of the same type. please trace the following relations.

[itex] P(X│A_1 A_2…A_n )= \frac{P(X)}{P(A_1 A_2…A_n )} P(A_1 A_2…A_n│X)= [/itex]

[itex] \frac{P(X)}{P(A_1 A_2…A_n )}[ (-1)^{n-1} P(⋃_{i=1}^{n} A_i │X)+(-1)^{n-2} {n \choose 1} P(A_1│X)+ [/itex]

[itex] (-1)^{n-3} {n \choose 2}P(A_1 A_2│X)+…+{n \choose n-1}P(A_1 A_2…A_{n-1}│X) ] [/itex]

[itex] = \frac{1}{P(A_1 A_2…A_n )} [ (-1)^{n-1} P(X|⋃_{i=1}^{n}A_i ) P(⋃_{i=1}^{n}A_i)+ (-1)^{n-2} {n \choose 1}P(X│A_1 )P(A_1 )+(-1)^{n-3} {n \choose 2} P(X│A_1 A_2 )P(A_1 A_2 )+ [/itex]

[itex] …+{n \choose n-1}P(X│A_1 A_2…A_{n-1} )P(A_1 A_2…A_{n-1}) ] [/itex]

[itex] =(-1)^{n-1} \frac{P(⋃_{i=1}^{n}A_i )}{P(A_1 A_2…A_n )} P(X|⋃_{i=1}^{n}A_i )

+(-1)^{n-2} \frac{P(A_1 )}{P(A_1 A_2…A_n )} {n \choose 1} P(X│A_1 )+(-1)^{n-3} \frac{P(A_1 A_2 )}{P(A_1 A_2…A_n )} {n \choose 2} P(X│A_1 A_2 )+[/itex] [itex]…+{n \choose n-1} \frac{P(A_1 A_2…A_{n-1})}{(P(A_1 A_2…A_n)} P(X│A_1 A_2…A_{n-1} ) [/itex]

Is everything OK? thanks a lot in advance for your involvement!

The problem:

Assume the events [itex] A_i [/itex] for [itex] i=1,2,...,n [/itex] are of the same type. please trace the following relations.

[itex] P(X│A_1 A_2…A_n )= \frac{P(X)}{P(A_1 A_2…A_n )} P(A_1 A_2…A_n│X)= [/itex]

[itex] \frac{P(X)}{P(A_1 A_2…A_n )}[ (-1)^{n-1} P(⋃_{i=1}^{n} A_i │X)+(-1)^{n-2} {n \choose 1} P(A_1│X)+ [/itex]

[itex] (-1)^{n-3} {n \choose 2}P(A_1 A_2│X)+…+{n \choose n-1}P(A_1 A_2…A_{n-1}│X) ] [/itex]

[itex] = \frac{1}{P(A_1 A_2…A_n )} [ (-1)^{n-1} P(X|⋃_{i=1}^{n}A_i ) P(⋃_{i=1}^{n}A_i)+ (-1)^{n-2} {n \choose 1}P(X│A_1 )P(A_1 )+(-1)^{n-3} {n \choose 2} P(X│A_1 A_2 )P(A_1 A_2 )+ [/itex]

[itex] …+{n \choose n-1}P(X│A_1 A_2…A_{n-1} )P(A_1 A_2…A_{n-1}) ] [/itex]

[itex] =(-1)^{n-1} \frac{P(⋃_{i=1}^{n}A_i )}{P(A_1 A_2…A_n )} P(X|⋃_{i=1}^{n}A_i )

+(-1)^{n-2} \frac{P(A_1 )}{P(A_1 A_2…A_n )} {n \choose 1} P(X│A_1 )+(-1)^{n-3} \frac{P(A_1 A_2 )}{P(A_1 A_2…A_n )} {n \choose 2} P(X│A_1 A_2 )+[/itex] [itex]…+{n \choose n-1} \frac{P(A_1 A_2…A_{n-1})}{(P(A_1 A_2…A_n)} P(X│A_1 A_2…A_{n-1} ) [/itex]

Is everything OK? thanks a lot in advance for your involvement!

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