2026 EJC P2 Q11

The masses of mangoes in kg have the distribution \(\mathrm N(0.3,0.1^2)\).

1. Find the probability that two randomly selected mangoes weigh at least 0.5 kg in total.[2]

Mark sells fruit boxes, each of which contain 10 randomly selected blueberries and 2 randomly selected mangoes. Let the number of unblemished blueberries in a randomly selected fruit box be \(U\). On average, the probability of a blueberry being unblemished is \(p\).

2. State a reason, in the context of the question, why \(U\) might not be well-modelled by a binomial distribution.[1]

Assume now that \(U\) has the distribution \(\mathrm B(10,p)\). The probability that a randomly selected fruit box contains at least 8 unblemished blueberries is 0.9.

3. Find the value of \(p\).[2]

Each fruit box is classified as grade A, B or C according to the following criteria:

4. Show that the probability that a randomly chosen fruit box is classified as grade C is 0.316, correct to 3 significant figures.[2]
5. Hence find the probability that a randomly selected grade C fruit box contains at least 8 unblemished blueberries.[3]
6. The selling prices of a grade A, grade B and grade C fruit box are \(\$12\), \(\$10\) and \(\$6\) respectively. Find the probability that the total selling price of 5 randomly selected fruit boxes is less than \(\$37\).[3]

Mark decides to amend the criteria for grade A boxes as follows:

• Each mango in the box weighs at least 0.3 kg and the box contains at least 9 unblemished blueberries.

7. State how the percentage of fruit boxes classified as grade A will change, and explain your answer.[2]