Question

In proof testing of circuit boards, the probability that a particular diode will fail is

0.015. Suppose a circuit board contains 180 diodes.

a) Consider the random variable “number of diodes that fail out of 180 diodes”.

State the original distribution and the approximating distribution along with the

parameters. Also, justify why we can use the approximating distribution.

b) Using the approximating distribution, find the probability that exactly 3 diodes will

fail.

c) Using the approximating distribution, find the probability that at least 3 diodes will

fail.

d) Each board works properly only if all the diodes work. If 8 boards are shipped to

a particular customer, then what is the probability that at least 6 boards will work

properly?

Answer 1

a)

original distribution is binomial wth paramter n=180 and p=0.015

approximating distribution is poisson with paramer =np=180*0.015=2.7

this approximation is justified as n>100 and p<0.1 and np<10

b)

probability that exactly 3 diodes will fail =P(X=3)=e^-2.7*2.7³/3! =0.2205

c)

probability that at least 3 diodes will fail=P(X:>=3)=1-P(X<=2)=1-(P(X=0)+P(X=1)+P(X=2))

=1-(e^-2.7*2.7⁰/0!+e^-2.7*2.7¹/1! +e^-2.7*2.7²/2!) =1-0.4936 =0.5064

d)P(a board works properly)=P(0 didode fail)=e^-2.7 =0.0672

probability that at least 6 boards will work properly =P(X>=6)=P(X=6)+P(X=7)+P(X=8)

=

=0.000002