S1 June 2010 Q4

EdexcelOld spec10 marksProbability

4. The Venn diagram in Figure 1 shows the number of students in a class who read any of 3 popular magazines \(A\), \(B\) and \(C\).

Venn diagram in a box: A only 4, A and B only 2, B only 5, B and C only 3, C only 6, outside 10; A and C do not overlap — Figure 1

One of these students is selected at random.

(a) Show that the probability that the student reads more than one magazine is \(\dfrac{1}{6}\). (2)

(b) Find the probability that the student reads \(A\) or \(B\) (or both). (2)

Given that the student reads at least one of the magazines,

(d) find the probability that the student reads \(C\). (2)

(e) Determine whether or not reading magazine \(B\) and reading magazine \(C\) are statistically independent. (3)

Mark scheme (a)

Scheme	Marks
\(\dfrac{2 + 3}{\text{their total}} = \dfrac{5}{\text{their total}} = \dfrac{1}{6}\) ( given answer)	M1 A1cso
	(2)

Notes

M1 for \(\dfrac{2 + 3}{\text{their total}}\) or \(\dfrac{5}{30}\)

Mark scheme (b)

Scheme	Marks
\(\dfrac{4 + 2 + 5 + 3}{\text{total}},\ = \dfrac{14}{30}\) or \(\dfrac{7}{15}\) or \(0.4\dot{6}\)	M1 A1
	(2)

Notes

M1 for adding at least 3 of “4, 2, 5, 3” and dividing by their total to give a probability
Can be written as separate fractions substituted into the completely correct Addition Rule

Scheme	Marks
\(\mathrm{P}(A \cap C) = 0\)	B1
	(1)

Mark scheme (d)

Scheme	Marks
\(\mathrm{P}(C \mid \text{reads at least one magazine}) = \dfrac{6 + 3}{20} = \dfrac{9}{20}\)	M1 A1
	(2)

Notes

M1 for a denominator of 20 or \(\dfrac{20}{30}\) leading to an answer with denominator of 20

\(\dfrac{9}{20}\) only, 2/2

Mark scheme (e)

Scheme	Marks
\(\mathrm{P}(B) = \dfrac{10}{30} = \dfrac{1}{3},\ \mathrm{P}(C) = \dfrac{9}{30} = \dfrac{3}{10},\ \mathrm{P}(B \cap C) = \dfrac{3}{30} = \dfrac{1}{10}\) or \(\mathrm{P}(B\|C) = \dfrac{3}{9}\)	M1
\(\mathrm{P}(B) \times \mathrm{P}(C) = \dfrac{1}{3} \times \dfrac{3}{10} = \dfrac{1}{10} = \mathrm{P}(B \cap C)\) or \(\mathrm{P}(B\|C) = \dfrac{3}{9} = \dfrac{1}{3} = \mathrm{P}(B)\)	M1
So yes they are statistically independent	A1cso
	(3)
	(10 marks)

Notes

1st M1 for attempting all the required probabilities for a suitable test
2nd M1 for use of a correct test - must have attempted all the correct probabilities. Equality can be implied in line 2.
A1 for fully correct test carried out with a comment