A company receives an order to produce 150 customized Widgets of which 2%on average happen to be faulty.

Question

a) What is the chance of fulfilling the order by producing only the required  number of widgerts

b)How many widget are expected to be good on average on this case?

c) how mant widgets need to be produced to have on average the required number of good widets?

d) what is the chance of fulfilling the order by producing this number of widget?
 

Answer 1

(a) The probability of one widget being good is 0.98. For all to be good this must happen 150 times, so the overall probability of good widgets is 0.98^150=0.048296 approximately or 4.83%.

(b) 98 out of 100 widgets on average should be good, so that's 147 out of 150.

(c) 0.98x=150 where x is the number of widgets needed to produce on average 150 good ones. So x=150/0.98=153 approx.

(d) We need no more than 3 faulty widgets in a batch of 153. (0.98+0.02)^153=1 gives the total distribution of good and bad widgets. If we expand the first four terms we get:

1st term: 153 good, (0.045456)

2nd term: 152 good, 1 bad, (0.141934)

3rd term: 151 good, 2 bad, (0.220142)

4th term: 150 good, 3 bad. (0.224634)

Adding these gives us the probability of at least 150 good widgets. The total comes to 63.37% approx.

If we start with 154 widgets and sum the first 5 terms of the expansion the probability of 150 good widgets rises to 80.33%.