(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%.
