Question

Suppose 21​% of the population are 64 or​ over, 27​% of those 64 or over have​ loans, and 52​% of those under 64 have loans. Find the probabilities that a person fits into the following categories. ​(a) 64 or over and has a loan ​(b) Has a loan ​(c) Are the events that a person is 64 or over and that the person has a loan​ independent? Explain.

Answer 1

a) 0.0567

b) 0.4717

c) 0.099057

Explanation:

Given:

P( population are 64 or​ over ) = 21% = 0.21

P( Population are under 64 ) = 1 - P( population are 64 or​ over )

= 1 - 0.21 = 0.79

P( 64 or over have​ loans ) = 27% = 0.27

P( under 64 have loans ) = 52% = 0.52

Now,

a) P( 64 or over and has a loan )

= P( 64 or over have​ loans ) × P( population are 64 or​ over )

= 0.27 × 0.21

= 0.0567

b) P( Has a loan )

= P( population are 64 or​ over ) × P( 64 or over have​ loans ) + P( Population are under 64 ) × P( under 64 have loans )

= ( 0.21 × 0.27 ) + ( 0.79 × 0.52 )

= 0.0609 + 0.4108

= 0.4717

c) P( events that person is 64 or over and that the person has a loan​ independent )

= P( population are 64 or​ over ) × P( Has a loan )

= 0.21 × 0.4717

= 0.099057