Suppose 20% of the population are 65 or over, 26% of those 65 or over have loans, and 53% of those under 65 have loans. Find the probabilities that a person fits into the following categories. (a) 65 or over and has a loan (b) Has a loan (c) Are the events that a person is 65 or over and that the person has a loan independent? Explain.

The probability that a person is 65 or over and has a loan is 0.052, the probability that a person has a loan is 0.476, and the probability that a person is 65 or over and that the person has a loan independent is 0.0952.

Given :

Suppose 20% of the population are 65 or over, 26% of those 65 or over have loans, and 53% of those under 65 have loans.

Let A represent the person who is 65 or more and B represents the person who has a loan.

If (P(A) = 20% = 0.2) then (P(A') = 1 - 0.2 = 0.8)

If (P(B|A) = 26% = 0.26) then (P(B'|A) = 1 - 0.26 = 0.74)

If (P(B|A') = 53% = 0.53) then (P(B'|A') = 1 - 0.53 = 0.47)

a) The probability that a person is 65 or over and has a loan is:

= 0.052

b) The probability that a person has a loan is:

        = 0.476

c) The probability that a person is 65 or over and that the person has a loan independent is:

= P(A).P(B)

= (0.2)(0.476)

= 0.0952

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(a)The probability that a person 65 or over and has a loan is 0.052.

(b)The probability that a person 65 or over and has a loan is 0.476

(c) The probability that a person is 65 or over and has a loan independent is 0.0952

Step-by-step explanation:

Conditional probability:

Let A and B any two event is connected to a given random experiment E. The conditional probability of the event A on the hypothesis that the event B is occurred, denoted by P(A|B), is defined as,

So, P(A∩B) = P(B).P(A|B)

Given that,

The age of 20% of population are 65 years or over.

26% of those whose age 65 years or more have loans.

53% of those whose age under 65 years have loans.

E= the person is 65 or more.

F= the person has loan.

P(E)= 20%=0.2

P(F|E)=26%=0.26

P(F|E')=53%=0.53

So,

P(E')=1-0.2=0.8

P(F'|E)= 1-0.26=0.74

P(F'|E')= 1-0.53=0.47

(a)

The probability that a person 65 or over and has a loan is

=P(E∩F)

=P(F|E).P(E)

=0.26×0.2

=0.052

(b)

A person who has a loan either 65 or more , or under 65.

P(F)= P(F∩E)+P(F∩E')

     =P(F∩E)+P(E').P(F|E')

     = 0.052+ (0.8).(0.53)

     =0.476

(c)

Independent event:

If A and B are two independent event,

Then,P(A∩B)=P(A).P(B)

The probability that a person is 65 or over and has a loan independent is

=P(E∩F)

=P(E).P(F)

=(0.2)×(0.476)

=0.0952  

step-by-step explanation:

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