Fifty percent of households say they would feel secure if they had? $50,000 in savings. You randomly select 8 households and ask them if they would feel secure if they had? $50,000 in savings. Find the probability that the number that say they would feel secure is? (a) exactly? five, (b) more than? five, and? (c) at most five.

 

a)= 0.21875

 

(or ans =0.2188 if round 4 decimal)

 

b) 0.14453125

(if round 4 decimal places then

ans =0.1445)

 

c) 0.85546875

(if round 4 decimal places then

ans = 0.8555)

Step-by-step explanation

It is case of Binomial distribution

 

Probability mass function of Binomial distribution -

 

P(X=r)= ?(rn?)prqn−r?

 

where

?(rn?)=(n−r)!r!n!??

 

 

Here

n= 8

 

probability of success

p=50%= 0.50

 

q= 1-p= 1-0.50= 0.5

 

a)

Need to find

P(X=5)= ?(58?)(0.50)5(0.50)8−5?

 

as

 

?(58?)=(8−5)!5!8!?=5!×3×2×18×7××6×5!?? = 56

 

so

P(X=5)= (56)(0.50)?5(0.50)3?

 

= 0.21875

(or it is =0.2188 if round 4 decimal)

 

 

b) more than 5

need to find

P(X>5)= P(X=6)+P(X=7)+P(X=8)

 

now

P(X=6)= ?(68?)(0.50)6(0.50)2?

as

?(68?)=(8−6)!6!8!?=6!×2×18×7×6!?? =,28

so

P(X=6)=(28)(0.5)?6(0.50)2?

=0.109375

 

similarly

P(X=7)= ?(78?)(0.5)7(0.5)1?

 

=(8)(0.5)?7(0.5)? = 0.03125

 

P(X=8)= ?88?(0.50)8(0.50)0?

 

=(1)(0.50)?8? = 0.00390625

 

so

P(X>5)= 0.109375+0.03125+

0.00390625

 

=0.14453125

or

=0.1445 (if round 4 decimal)

 

 

c) at most 5

neeed to find

P(X?≤? 5) =1-P(X>5)

 

As

P(X>5)= 0.14453125 (as done above)

 

so

P(X?≤? 5)= 1-0.14453125

= 0.85546875

or

= 0.8555 (if round 4 decimal)