Question

An office has a stock of identical printed forms which are used independently. On any working day, at most one of the forms is used, and the probability that one form is used is 1/3 . There are 250 working days in the year.

(i) Using a suitable approximation, calculate the number of forms that must be in stock at the beginning of the year if there is to be a 95% probability that they will not all be used before the end of the year. [5 marks]

(ii) If one form in one hundred is unusable due to faulty printing and these faults occur at random, calculate the probability that in a batch of 250 forms there will be not more than one which is unusable.

Answer 1

(i) A suitable approximation for the binomial distribution is the normal distribution with the same mean and the standard deviation as the original binomial distribution. Here, we have:

We now use a normal distribution. A 95% probability corresponds to a z-score of 1.64. So, the number of forms x that give this z-score is given by:

Answer: There must be a stock of at least 96 forms to have a probability of 95% of not using all till the year end.

(ii) We again model this using a normal distribution.

For 1 or less forms, we take . Note that this is the middle number between 1 (which we want) and 2 (which we don't want). So, we get the z-score as:

Using a z-table, we get the corresponding probability as 0.2611. This is the probability that there will be less than 1.5 forms. So, this is the probability that there will be 0 or 1 unusable forms.

Answer: Probability = 0.2611