20. if (A∩B∩C)≠0 and P(C|A∩B)=P(C|B) , show that : P(A|B∩C)=P(A|B)

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21 . suppose that A,B and c are independent events .show that
i. A and (A∩B) are independent.
ii. A and (AUB) are independent.
iii. A^c and (A∩c^c) are independent.

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22. in a city ,25% of the cars emit pollutions to the air of the city .if a car failed to pass the emission of air pollutants with probability 0.99 and if a car that is not emitting air pollutants also fails the test with probability 0.17 .what is the probability that a car that fails the test actually emits air pollution ?

2. Originally Posted by compufatwa
20. if (A∩B∩C)≠0 and P(C|A∩B)=P(C|B) , show that : P(A|B∩C)=P(A|B)
$P(A|B \cap C)=\frac{P(A \cap B \cap C)}{P(B \cap C)}=\frac{P(C \cap A \cap B)}{P(B \cap C)} = \frac{P(C|A \cap B)P(A \cap B)}{P(B \cap C)}$
$= \frac{P(C|B)P(A \cap B)}{P(B \cap C)} = \frac{\frac{P(C \cap B)}{P(B)}P(A \cap B)}{P(B \cap C)} = \frac{P(A \cap B)}{P(B)} = P(A|B)$
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Originally Posted by compufatwa
21 . suppose that A,B and c are independent events .show that
i. A and (A∩B) are independent.
ii. A and (AUB) are independent.
iii. A^c and (A∩c^c) are independent.
To show independence, we need to show that
$P(A \cap B) = P(A)P(B)$
so,
i) $\, P(A \cap (A \cap B)) = P((A \cap A) \cap B) = P(A \cap B)$
$= P(A)P(B) = P(A \cap A)P(B) = P(A)P(A)P(B) = P(A)P(A \cap B)$
im not sure with it (i.e. if A is independent to itsefl)..
just do the same thing with the others
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