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.