Question

02) a) A department store manager has monitored the number of complaints received per week

about poor service. The probabilities for numbers of complaints in a week, established by this

review, are shown in the following table.

Number of

complaints

0 1 to 3 4 to 6 7 to 9 10 to 12 More

than 12

Probability 0.14 0.39 0.23 0.15 0.06 0.03

Let A be the event “there will be at least one complaint in a week” and B the event “there will be

fewer than ten complaints in a week.”

I. Find the probability of event A.

II. Find the probability of event B.

III. Find the probability of the complement of event A.

IV. Find the probability of the intersection of events A and B.

V. Are A and B mutually exclusive?

Answer 1

Solution:

I. P(A)=P(X $\ge$ 1)=1- P(X=0)=1-0.14=0.86

II. P(B)=P(X<10)=1-P(X$\ge$ 10)=1-P(X>12)-P(10$\le$X$\le$12)=1-0.03-0.06=0.91

III. P($\overline{A}$ )=P(X<1)=P(X=0)=0.14

IV. P(A$\bigcap$B)=P(1$\le$ X<10)=P(1$\le$ X$\le$ 3)+ P(4$\le$X$\le$6)+P(7$\le$X $\le$ 9)=

=0.39+0.23+0.15=0.77

V. The intersection of events A and B сontains elementary events X=1, X=2,..., X=9 and therefore, is not empty.

So, events A and B are not mutually exclusive.