In each batch of pralines that a chocolate factory produces, they are packed into boxes of $16$. D

Question

In each batch of pralines that a chocolate factory produces, they are packed into boxes of $16$. Due to the new health guidelines regarding sugar content, quality control tests are implemented and show that on average, the proportion of pralines that exceed the recommended sugar content is $p$. The number of pralines in a box that exceed the recommended sugar content is denoted by $X$. State, in context of the question, two assumptions needed for $X$ to be well-modelled by a binomial distribution.[2] State, in context of the question, two assumptions needed for $X$ to be well-modelled by a binomial distribution.[2] Assume now that $X$ is modelled by a binomial distribution. It is known that the most probable number of pralines that exceed the recommended sugar content is $2$. Find the exact range of values that $p$ can take.[3] It is known that the most probable number of pralines that exceed the recommended sugar content is $2$. Find the exact range of values that $p$ can take.[3] For each batch of pralines produced, the chocolate factory implements the following quality control system by testing the first random sample of $16$ pralines in a box: If there are at most $1$ praline that exceed the recommended sugar content in a box, the batch is accepted. If there are $3$ or more pralines that exceed the recommended sugar content in a box, the batch is rejected. If there are $2$ pralines that exceed the recommended sugar content in a box, a second box of $16$ pralines is tested. The batch will be accepted if there are at most $1$ praline that exceed the recommended sugar content in the second box, otherwise the batch will be rejected. It is given that $p=0.15$. Find the probability that the batch of pralines is accepted under this quality control system.[2] Find the probability that the batch of pralines is accepted under this quality control system.[2] Given that the batch of pralines is accepted, find the probability that a second box of pralines is tested.[2]

Accepted Answer

(a)(i)The probability of pralines that exceed the recommended sugar content is constant at p for every praline. Whether a praline has exceeded the recommended sugar content is independent of any other pralines. (a)(ii) $\frac{2}{17}$$<$$p<\frac{3}{17}$ (b)(i)Required probability = 0.363 (3 sf) (b)(ii) Required probability = 0.217 (3 sf)

2025 CJC P2 Q8

2025 CJC P2 Q8