Introduction to Probability and Statistics for Engineers and Scientists

Problems 81

6. LetE,F,Gbe three events. Find expressions for the events that ofE,F,G

(a) onlyEoccurs;

(b) bothEandGbut notFoccur;

(c) at least one of the events occurs;

(d) at least two of the events occur;

(e) all three occur;

(f ) none of the events occurs;

(g) at most one of them occurs;

(h) at most two of them occur;

(i) exactly two of them occur;

(j)at most three of them occur.

7. Find simple expressions for the events

(a) E∪Ec;

(b) EEc;

(c) (E∪F)(E∪Fc);

(d) (E∪F)(Ec∪F)E∪Fc);

(e)(E∪F)(F∪G).

8. Use Venn diagrams (or any other method) to show that

(a) EF⊂E,E⊂E∪F;

(b) ifE⊂FthenFc⊂Ec;

(c) the commutative laws are valid;

(d) the associative laws are valid;

(e) F=FE∪FEc;

(f ) E∪F=E∪EcF;

(g) DeMorgan’s laws are valid.

9. For the following Venn diagram, describe in terms ofE,F, andGthe events
   denoted in the diagram by the Roman numerals I through VII.

EF

G

S

I

II

IV

III

VII V

VI