Question

Seventy 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,

(c) at most five.

Answer 1

Define random variable X: number of households feel secure if they have $50000 in savings.

X follows binomial distribution with n = 8 and p = 0.70

If x follows binomial distribution with n and p then,

                   x = 0,1,2...............,n

where                            n! = 1 * 2 * 3 * 4*.......*n

a) Here we have to find P(X = 5)

                     

                     

                    

                    = 56 * 0.16807 * 0.027

                    = 0.2541                 (Round to 4 decimal)

The probability that the number that say households would feel secure is exactly 5 is 0.2541

b) Here we have to find P(X > 5)

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

                     

                     

                                          ( 2! = 1 * 2 = 2)

                    = 28 * 0.117649 * 0.09

                    = 0.2965                 (Round to 4 decimal)

                     

                     

                                         (1! = 1)

                    = 8 * 0.082354 * 0.30

                    = 0.1977              (Round to 4 decimal)

                                                  

                     

                      = 0.0576                 (Round to 4 decimal)

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

              = 0.2965 + 0.1977 + 0.0576

              = 0.5518

The probability that the number that say households would feel secure is more than 5 is 0.5518

c) Here we have to find

   = 1 - 0.5518 (From part b)

   = 0.4482

The probability that the number that say households would feel secure is at most 5 is 0.4482