Q. 2.6 - A First Course in Probability

Question

Let E, F, and G be three events. Find expressions for the events so that, of E, F, and G,

(a) only E occurs;

(b) both E and G, but not F, occur;

(c) at least one of the events occurs;

(d) at least two of the events occur;

(e) all three events occur;

(f) none of the events occurs;

(g) at most one of the events occurs;

(h) at most two of the events occur;

(i) exactly two of the events occur;

(j) at most three of the events occur.

Step-by-Step Solution

Answer

The events are as follows

(a) $E\overline{FG}$

(b) $E\overline{F}G$

(c) $E\cup F\cup G$

(d) $EF\cup EG\cup FG$

(e) $EFG$

(f) $\overline{EFG}$

(g)  $EFG\cup E\overline{FG}\cup \overline{E}F\overline{G}\cup E\overline{FG}$

(h) $\overline{\left(EFG\right)}$

(i) $EF\overline{G}+E\overline{F}G+EF\overline{G}$

(j) $S$

Step 1 : Given information

Three events $E, F \mathrm{and} G$

Step 2 : Part(a)

Only E occours$=E\overline{F}\overline{G}$

Step 3 : Part(b)

Both E and G occors but not F$=E\overline{F}G$

Step 4 : Part(c)

Atleast one of the events occor $=E\cup F\cup G$

Step 5 : Part(d)

Atleast two of the events occor$=EF\cup FG\cup EG$

Step 6 : Part(e)

All three events occor$=EFG$

Step 7 : Part(f)

None of the events occor$=\overline{EFG}$

Step 8 : Part(g)

Atmost one of the two events occor$=\overline{EFG}\cup E\overline{FG}\cup \overline{E}F\overline{G}\cup E\overline{FG}$

Step 9 : Part(h)

Atmost two of the events occor$=\overline{\left(EFG\right)}$

Step 10 : Part(i)

Exactly two of the events occor$=EF\overline{G}+E\overline{F}G+EF\overline{G}$

Step 11 : Part(j)

Atmost three of the events occor$=S$

where S denotes a super-set of all the sets possible.