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.

\(X \sim {\text{Po}}(1.2)\)

\({\text{P}}(X = 3) \times {\text{P}}(X = 0)\)     (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]

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.