A factory manufactures a large number of erasers in a variety of colours. Each box of erasers contai

Question

A factory manufactures a large number of erasers in a variety of colours. Each box of erasers contains $36$ randomly chosen erasers. On average, $20$% of erasers in the box are blue. State, in context, two assumptions needed for the number of blue erasers in a box to be well modelled by a binomial distribution. [2] Find the probability that a randomly chosen box of erasers contain at most six blue erasers. [1] $200$ randomly chosen boxes are packed into a carton. A carton is considered acceptable if at least $40$% of the boxes contain at most six blue erasers each. Find the probability that a randomly chosen carton is acceptable. [3] The cartons are exported by sea. Over a one-year period, there are $30$ shipments of $150$ cartons each. Using a suitable approximation, find the probability that the mean number of acceptable cartons per shipment for the year is less than $80$. The owner decided to change the proportion of blue erasers to $p$. A box of erasers is chosen. Write down in terms of $p$, the probability that the box contains exactly one blue eraser. [1] The probability that a box contains exactly one blue eraser is twice the probability that the box contains exactly two blue erasers. Write an equation in terms of $p$, and hence find the value of $p$. [2]

Accepted Answer

$p = \frac{1}{36}$

2019 SAJC P2 Q10

2019 SAJC P2 Q10

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