Question

(a)          In a study into consumer behaviour and service dissatisfaction, it is found that the probability of event A, that a consumer will switch their mobile phone service provider, is 0.6, and the probability of event B, that a consumer will switch their home electricity supply provider, is 0.5. The probability that a consumer will switch their mobile phone service provider and will switch their home electricity supply provider, P(A ∩ B), is found to be 0.42.

1. What is the probability that a consumer does not switch their mobile phone service provider? i.e. Find P(A).

2. What is the probability that the consumer will switch their home electricity supply provider given the consumer switches their mobile phone service provider? i.e. Find P(B|A).

3. Are the events A and B mutually exclusive? Give a reason for your answer using probabilities to demonstrate.

4. Are the events A and B independent? Give a reason for your answer using probabilities to demonstrate.

(b)          A company is contracted to install water meters in Irish residential properties. Say the company believes that 12 % of all Irish homeowners are opposed to installation of a water meter in their home. Suppose that a residential estate consists of 20 houses. Let X be the variable the number out of 20 homeowners opposed to water meter installation. Assuming the variable X follows a binomial distribution,

(i) What is the value of, E(X), the expected number of homeowners that would be opposed to water meter installation?

(ii) What is the probability that exactly 2 of the 20 homeowners are opposed to water meter installation?

(iii) What is the probability that none of the 20 homeowners are opposed to water meter installation?

(iv) What is the probability that at least one of the 20 homeowners are opposed to water meter installation?

Answer 1

(a)

P(A)=0.6

P(B)=0.5

(1)

=0.4

(2)  

=0.42/0.6

=0.7

(3) The 2 events are said to be mutually exclusive when the 2 can not happen simultaneously. That is their intersection must be 0. here,

is non zero

so the 2 events A, B are not mutually exclusive .

(4) The two events are said to be independent  if:

  

0.42=0.6*0.5

0.42=0.3

Clearly  the two quantities are not equal

A,B not independent.

(b)

(1) P(opposing the water mt installation)=0.12

total no of houses=20

mean of binomial=np

=20*0.12

mean =2.4

(2)   

(3) P(all the 20 homes will oppose to water mt installation)=P(X=20)

P(none of will oppose)=1-P(X=20)

=

=1-(0.12^20)

(4)

=

=