A shop sells apples, pears and peaches. The weights, in grams, of these three types of fruit may be assumed to be normally distributed with means and standard deviations as given in the following table.

Alan buys 1 apple and 1 pear while Brian buys 1 peach. Calculate the probability that the combined weight of Alan’s apple and pear is greater than twice the weight of Brian’s peach.

let X, Y, Z denote respectively the weights, in grams, of a randomly chosen apple, pear, peach

then \(U = X + Y - 2Z{\text{ is N}}(115 + 110 - 2 \times 105,{\text{ }}{5^2} + {4^2} + {2^2} \times {3^2})\)     (M1)(A1)(A1)

Note: Award M1 for attempted use of U.

i.e. N(15, 77)     A1

we require

\({\text{P}}(X + Y > 2Z) = {\text{P}}(U > 0)\)     M1A1

\( = 0.956\)     A2

Note: Award M0A0A2 for 0.956 only.

[8 marks]

Solutions to this question again illustrated the fact that many candidates are unable to distinguish between nX and \(\sum\limits_{i = 1}^n {{X_i}} \) so that many candidates obtained an incorrect variance to evaluate the final probability.