(a)     Find the probability that the weight of a randomly chosen apple is more than double the weight of a randomly chosen pear.

(b)     A shopper buys 3 apples and 4 pears. Find the probability that the total weight is greater than 1000 grams.

(a)     Let X, Y (grams) denote respectively the weights of a randomly chosen apple, pear.

Then

$X - 2Y{\text{ is N}}(200 - 2 \times 120,{\text{ }}{15^2} + 4 \times {10^2}),$     (M1)(A1)(A1)

i.e. ${\text{N}}( - 40,{\text{ }}{25^2})$     A1

We require

${\text{P}}(X > 2Y) = {\text{P}}(X - 2Y > 0)$     (M1)(A1)

$= 0.0548$     A2

[8 marks]

(b)     Let $T = {X_1} + {X_2} + {X_3} + {Y_1} + {Y_2} + {Y_3} + {Y_4}$ (grams) denote the total weight.

Then

$T{\text{ is N}}(3 \times 200 + 4 \times 120,{\text{ }}3 \times {15^2} + 4 \times {10^2}),$     (M1)(A1)(A1)

i.e. ${\text{N(1080, 1075)}}$     A1

${\text{P}}(T > 1000) = 0.993$     A2

[6 marks]

Total [14 marks]

The response to this question was disappointing. Many candidates are unable to differentiate between quantities such as $3X{\text{ and }}{X_1} + {X_2} + {X_3}$ . While this has no effect on the mean, there is a significant difference between the variances of these two random variables.