Answer to Question #249137 in Finance for Jenelle

Question

a) Two events, M and N, are such that P(M) = 0.6, P(M ∩N) = 0.2, P(M ∪N) = 0.85

Find

(i) P(N |M) [2]

State, with reason, whether M and N are

(ii) mutually exclusive [2]

(iii) independent. [2]

b) The following table shows the results of a sample survey carried out by a fast-food restaurant to determine whether persons had noticed an improvement in its service in the last month

Voting Results

Sex

Improvement Seen

No improvement

No Opinion

Male

25

21

34

Female

16

38

16

i. How many people were in the survey? [1]

ii. A patron was selected at random from the sample.

a) Determine the probability that the patron thought that there was an improvement in service. [2]

b) Determine the probability that the patron was a female or thought that there was no improvement in the service. [3]

c) Given that the patron thought there was an improvement, what is the probability that the patron was a male?

Answer 1

a) (i) $P(N|M)=\frac{P(N\cap M}{P(M)}=0.2/0.6=0.3333$

ii) For event M and N to be mutually exclusive, we should have $P(M\cap N)=0$

$P(M \cap N)=0.2 \ne0$ , so M and N are not mutually exclusive.

iii) For event M and N to be independent, we should have $P(M \cap N )=P(M)P(N)$

$P(M \cup N)=P(M)+P(N)-P(M \cap N)$

$0.85=0.6+P(N)-0.2$

$P(N)=0.45$

So, $P(M)P(N)=(0.6)(0.45)=0.27\ne P(M\cap N).$ So, M and N are not independent.

b) i) Total people =25+21+34+16+38+16=150

ii) (a) P(improvement in service) = (female +male (improvement seen))$/$ (total people)=(16+25)$/150=41/150$

b) P(female (no improvement) = $\frac{P(female)}{total people}+\frac{P(no-improvement)}{total people}-\frac{P(female\space and\space no\space improvement)}{total people}=(16+38+16)/150+(21+38)/150-38/150=70/150+59/150-38/150=0.6067$

c) P(male improvement) =P( male and improvement)$/$ (P(improvement)=$25/(25+16)=0.6098$