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# Solving Probability Theory Questions

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Grindit & Floggit Ltd. make customized Widgets at a cost of \$150 per unit of which 3% (on average) happen to be out of tolerance. Up to 115 Widgets can be produces in one continuous run whose set up incurs an additional cost of \$10 000. Since manufacturing runs are set to produce one of many variants of Widget, Grindit & Floggit estimate that Widgets made in excess of a firm order have no residual value. All Widgets are tested to see if they are within tolerance only after a manufacturing run has been complete.

Grindit & Floggit receive an order for 100 Widgets
a) What is the chance to fulfill the order by producing exactly 100 Widgets?
b) What is the chance to fulfill the order by producing 105 Widgets?
c) How many Widgets should be produced to fulfill the order with less than 5% risk?
d) What is the cost of production of each of 100 resulting Widgets in the last case? What price do you think Grindit & Floggit should be charging the customer per Widget in the last case if they want to make 10% profit (round to nearest penny)?

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The expert solves probability theory.

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a) The problem can be modeled with a binomial distribution where we consider "Success" to be that a Widget is in tolerance.
Let Success = the widget is within tolerance
Let Failure = the widget is out of tolerance
Let p = the probability of success = .07
Let n = the number of widgets produced in a single run
Let X = the number of Widgets that are within tolerance

X is a binomial random variable with parameters and .

The probability that X assumes a particular value can be found by using the binomial density function:

Using this function with and we get:

Note: You can easily calculate Binomial ...

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