S1 June 2012 Q4

EdexcelOld spec9 marksProbability

Figure 1: Venn diagram with B only 4, B and T 5, T only 3, W 7 (separate), outside 6 — Figure 1

Figure 1 shows how 25 people travelled to work.

Their travel to work is represented by the events

\(B\)    bicycle
\(T\)    train
\(W\)    walk

(a) Write down 2 of these events that are mutually exclusive. Give a reason for your answer. (2)

(b) Determine whether or not \(B\) and \(T\) are independent events. (3)

One person is chosen at random.

Find the probability that this person

(d) travels to work by bicycle and train. (1)

(e) Given that this person travels to work by bicycle, find the probability that they will also take the train. (2)

Mark scheme (a)

Scheme	Marks
\(B, W\) or \(T, W\) [ accept \(B \cup T, W\) or \(B \cap T, W\) ] [Condone \(\mathrm{P}(B), \mathrm{P}(W)\) etc]	B1
Since there is no overlap between the events or cannot happen together (o.e.) (Accept comment in context e.g. “no one walks and takes the train”)	B1
	(2)

Notes

1st B1 for a suitable pair. Do not accept universally exclusive pairs such as \(B\) and \(B^{\prime}\) etc

2nd B1 for any correct statement. Accept use of symbols e.g.: \(B \cap W = \varnothing\) or \(\mathrm{P}(T \cap W) = 0\) etc. But \(T \cap W = 0\) is B0 (since it is not a correct statement)

Mark scheme (b)

Scheme	Marks
e.g. \(\mathrm{P}(B) = \dfrac{9}{25},\ \mathrm{P}(T) = \dfrac{8}{25},\ \mathrm{P}(B \cap T) = \dfrac{5}{25}\)	M1
\(\mathrm{P}(B \cap T) \ne \mathrm{P}(B)\times\mathrm{P}(T)\) \([0.2 \ne 0.36\times 0.32 = 0.1152\) o.e.]	M1
So \(B\) and \(T\) are not independent	A1cso
	(3)

Notes

1st M1 for an attempt at all required probabilities with labels for a suitable test (allow one error). Accept use of \(A\) and \(B\) as long as they can be identified as \(B\) and \(T\) by correct probabilities. Must be probabilities not integers such as 5, 9, 8 etc for both these M marks

2nd M1 for \(\mathrm{P}(B)\times\mathrm{P}(T)\) evaluated (correct for their probabilities)
or \(\mathrm{P}(B \cap T) \ne \mathrm{P}(B)\times\mathrm{P}(T)\) stated or implied in symbols or using their probabilities.
or \(\mathrm{P}(B \mid T) \ne \mathrm{P}(B)\) or \(\mathrm{P}(T \mid B) \ne \mathrm{P}(T)\) stated or implied in symbols or using their probabilities.

A1 for a conclusion of not independent. Requires all probabilities used to be correct and seen. This A mark is dependent on both Ms

NB \(\mathrm{P}(B \mid T) = \dfrac{5}{8}\) & \(\mathrm{P}(B) = \dfrac{9}{25}\) or \(\mathrm{P}(T \mid B) = \dfrac{5}{9}\) & \(\mathrm{P}(T) = \dfrac{8}{25}\) seen, followed by a correct conclusion scores 3/3

Scheme	Marks
\([\mathrm{P}(W) =]\ \dfrac{7}{25}\) or 0.28	B1
	(1)

Scheme	Marks
\([\mathrm{P}(B \cap T) =]\ \dfrac{5}{25}\) or \(\dfrac{1}{5}\) or 0.2	B1
	(1)

Mark scheme (e)

Scheme	Marks
\([\mathrm{P}(T \mid B) =]\ \dfrac{\mathrm{P}(T \cap B)}{\mathrm{P}(B)} = \dfrac{\text{"(d)"}}{(5 + 4)/25}\)	M1
\(= \dfrac{5}{9}\) or \(0.\dot{5}\)	A1
	(2)
	(9 marks)

Notes

M1 for a correct ratio of probabilities e.g. \(\dfrac{5/25}{(5 + 4)/25}\) or \(\dfrac{5}{5 + 4}\) or
A correct ratio expression and at least one correct (or correct f.t.) probability substituted.

A1 for \(\dfrac{5}{9}\) with no incorrect working seen but \(\dfrac{5}{9}\) following from \(\mathrm{P}(B \mid T)\) is 0/2. \(\dfrac{5}{9}\) alone is 2/2