A fruit seller sources his oranges from a supplier with a large plantation where $1$% of the orang

Question

A fruit seller sources his oranges from a supplier with a large plantation where $1$% of the oranges are rotten. The supplier picks the oranges regardless of whether the oranges are rotten or not, and packs $20$ oranges in each box at random. The random variable $X$ denotes the number of rotten oranges in a randomly chosen box. State in context two assumptions for $X$ to follow a binomial distribution.[2] Assume that $X$ follows a binomial distribution for the rest of this question. The fruit seller considers a box containing between $2$ and $5$ rotten oranges inclusive as substandard. Find the probability that a randomly chosen box is substandard.[1] Find the expected number of rotten oranges in a box.[1] A truckload of $10$ randomly chosen boxes of oranges is then sent to the fruit seller. Find the probability that not more than $2$ boxes from the entire truckload are substandard.[2] Upon receiving the oranges, the fruit seller inspects the boxes with these conditions: Accept the entire truckload if the first box opened has no rotten oranges. Reject the entire truckload if the first box opened has more than $1$ rotten orange. If the first box has exactly $1$ rotten orange, the fruit seller opens a second box. If the second box has at most $1$ rotten orange, he will accept the entire truckload. Otherwise, he will reject the entire truckload. Find the probability that he will reject the entire truckload.[2]

Accepted Answer

(a) The probability that an orange is rotten is a constant of 0.01. Whether an orange is rotten is independent of whether any other oranges are rotten. (b)$P(2\leq X\leq5)=0.0169$ (c)$E(X)=0.2$ (d)$P(Y\leq 2)=0.999\,\mathrm (3 s.f)$ (e)$P(A)=0.0196$

2025 ACJC P2 Q9

2025 ACJC P2 Q9