Answer to Question #234586 in Statistics and Probability for Milika

Question

A certain mobile money agency is an agent for Zanaco Express, Airtel, MTN and zamtel mobile money. Experience has shown that the probabilities of finding a float for withdrawals of amounts k5000 and above are given as 0.4, 0.5, 0.3 and 0.2, respectively. Experience has also shown that among his clients the possibilities that a client requires a service from Zanaco Express, Airtel, MTN and zamtel mobile money are given as 0.3, 0.4, 0.2, and 0.1, respectively. Assume that each client visits the agency for withdrawals from only one at a time
I) what is the probability that a client who wishes to withdraw k5000 will not be successful?
ii) what is the probability that an MTN client who wishes to withdraw k5000 will not be successful?
iii) what is the probability that an Airtel client who wishes to withdraw k5000 will not be successful?
iv) a certain client has just failed to make a withdraw of k7000 what is the probability that he/ she is a zamtel mobile money client?

Answer 1

Consider the following events:

D: withdraw k5000 will not be successful,

ZE: a service from Zanaco Express,

A: a service from Airtel,

MTN: a service from MTN,

ZMM: a service from zamtel mobile money.

i) By the Law of Total Probability

$$P(D)=P(ZE)P(D|ZE)+P(A)P(D|A)$$

$$+P(MTN)P(D|MTN)+P(ZMM)P(D|ZMM)$$

$$=0.3(1-0.4)+0.4(1-0.5)+0.2(1-0.3)$$

$$+0.1(1-0.2)=0.6$$

ii)

$$P(D|MTN)=1-0.3=0.7$$

iii)

$$P(D|A)=1-0.5=0.5$$

iv) By Bayes' Theorem

$$P(ZMM|D)=\dfrac{P(D|ZMM)P(ZMM)}{P(D)}$$

$$=\dfrac{(1-0.2)(0.1)}{0.6}=\dfrac{2}{15}\approx0.1333$$