**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 .**