Date | May 2013 | Marks available | 2 | Reference code | 13M.2.hl.TZ2.9 |
Level | HL only | Paper | 2 | Time zone | TZ2 |
Command term | Find | Question number | 9 | Adapted from | N/A |
Question
A small car hire company has two cars. Each car can be hired for one whole day at a time. The rental charge is US$60 per car per day. The number of requests to hire a car for one whole day may be modelled by a Poisson distribution with mean 1.2.
Find the probability that on a particular weekend, three requests are received on Saturday and none are received on Sunday.
Over a weekend of two days, it is given that a total of three requests are received.
Find the expected total rental income for the weekend.
Markscheme
\(X \sim {\text{Po}}(1.2)\)
\({\text{P}}(X = 3) \times {\text{P}}(X = 0)\) (M1)
\( = 0.0867 \ldots \times 0.3011 \ldots \)
\( = 0.0261\) A1
[2 marks]
Three requests over two days can occur as (3, 0), (0, 3), (2, 1) or (1, 2). R1
using conditional probability, for example
\(\frac{{{\text{P}}(3,{\text{ }}0)}}{{{\text{P}}(3{\text{ requests, }}m = 2.4)}} = 0.125{\text{ or }}\frac{{{\text{P}}(2,{\text{ }}1)}}{{{\text{P}}(3{\text{ requests, }}m = 2.4)}} = 0.375\) M1A1
expected income is
\(2 \times 0.125 \times {\text{US}}\$ 120 + 2 \times 0.375 \times {\text{US}}\$ 180\) M1
Note: Award M1 for attempting to find the expected income including both (3, 0) and (2, 1) cases.
\( = {\text{US}}\$ 30 + {\text{US}}\$ 135\)
\( = {\text{US}}\$ 165\) A1
[5 marks]
Examiners report
Part (a) was generally well done although a number of candidates added the two probabilities rather than multiplying the two probabilities. A number of candidates specified the required probability correct to two significant figures only.
Part (b) challenged most candidates with only a few candidates able to correctly employ a conditional probability argument.