1. ## events

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. Ac and (A∩cc) are independent.

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 ?

23.the number of employees working in factories A,B and C are , respectively 50,75 and 100 . suppose that of those employees ,50% ,60 % and 70%,respectively ,are women .assuming that the probability of resignation is the same for men and women and that woman resigned ,what is the probability that she is from factory c ?

24 . two balanced dice are rolled .if you know that the upper faces show different numbers ,what is the probability that one of the two faces show 4 .

2. We know that from independence that $\displaystyle P(AB) = P(A)P(B)$.
Look at
$\displaystyle \begin{array}{rcl} P(AB^c ) & = & P(A) - P(AB) \\ & = & P(A) - P(A)P(B) \\ & = & P(A)\left( {1 - P(B)} \right) \\ & = & P(A)P(B^c ) \\ \end{array}$
That shows that $\displaystyle A\,\& \,B^c$ are independent.

By changing position $\displaystyle B^c\,\& \,A$, we see $\displaystyle A^c\,\& \,B^c$ are independent.

You do the other.