2020 YIJC P2 Q8
A manufacturing company produces surgical masks. The surgical masks are randomly packed into boxes of $50$. On average, $15\%$ of the surgical masks are defective. The number of surgical masks that are defective in a box of $50$ pieces is denoted by $X$.
(i)
State, in context, two assumptions needed for the number of defective surgical masks in a box to be well modelled by a binomial distribution.
[2]
Assume now that the number of defective surgical masks in a box has a binomial distribution.
In a randomly chosen box of $50$ surgical masks, find the probability that there are
(ii)
more than $8$ defective surgical masks,
[2]
(iii)
at most $12$ defective surgical masks given that there are more than $8$ defective surgical masks.
[3]
For shipping purposes, the boxes are packed into cartons, with each carton containing $24$ boxes.
(iv)
Find the probability that, in a randomly chosen carton, there are at least $10$ boxes containing at most $8$ defective surgical masks.
[2]
The company also manufactures reusable masks which are packed into packets of $10$. The probability that a reusable mask is defective is $p$. It is known that the modal number of defective reusable masks in a packet is $1$.
(v)
Use this information to find exactly the range of values that $p$ can take.
[3]
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