Answer to Question #185313 in Statistics and Probability for Mariam

Question

A zoom be 999missile protection system consists of n radar sets operating independently, each with a

probability of .9 of detecting a missile entering a zone that is covered by all of the units.

(a) If n = 5 and a missile enters the zone, what is the probability that exactly four sets detect the

missile? At least one set?

(b) How large must n be if we require that the probability of detecting a missile that enters the

zone be .999?

8. Ten motors are packaged for sale in a certain warehouse. The motors sell for $100 each, but a zoom be 999.

Answer 1

(a).Let Y be the number of sets that detected the missile. Then, Y has a binomial distribution with n=5 and p=0.9.The probability that exactly four sets detected the missile is given as-

 

     $P(Y=4)=^5C_4\times (0.9)^4\times (0.1)^1=0.32805$

   And The probability that at least one set detected the missile is given as-

     $P(Y\ge 1)=1-P(Y=0)=1-^5C_0.(0.9)^0.(0.1)^5=1-0.00001=0.9999$

 (b). Evaluating for a few values of n, we obtain

          at n=2, p=0.99

            n=3, p=0.999

            n=4, p=0.9999

   Therefore for probability of 0.999, The value of n must be at least 3.

8. when There is a defect, you return the amount paid ( $100) and double the amount, which means that in total you will lose $100.

   The expected total cost of 1 motor is the sum of the products of the gain and their probability:

   

       $EV=100\times 0.92+(-100)\times 0.08=84$

    The expected net gain for the seller on the 10 motors is then:

         $10\times 84=840$