Date | May 2010 | Marks available | 8 | Reference code | 10M.3sp.hl.TZ0.4 |
Level | HL only | Paper | Paper 3 Statistics and probability | Time zone | TZ0 |
Command term | Calculate | Question number | 4 | Adapted from | N/A |
Question
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.
Markscheme
let X, Y, Z denote respectively the weights, in grams, of a randomly chosen apple, pear, peach
then U=X+Y−2Z is N(115+110−2×105, 52+42+22×32)U=X+Y−2Z is N(115+110−2×105, 52+42+22×32) (M1)(A1)(A1)
Note: Award M1 for attempted use of U.
i.e. N(15, 77) A1
we require
P(X+Y>2Z)=P(U>0)P(X+Y>2Z)=P(U>0) M1A1
=0.956=0.956 A2
Note: Award M0A0A2 for 0.956 only.
[8 marks]
Examiners report
Solutions to this question again illustrated the fact that many candidates are unable to distinguish between nX and n∑i=1Xin∑i=1Xi so that many candidates obtained an incorrect variance to evaluate the final probability.