S1 June 2014 (R) Q4

EdexcelOld spec9 marksProbability

4. \(A\) and \(B\) are two events such that

\[\mathrm{P}(B) = \frac{1}{2} \qquad \mathrm{P}(A \mid B) = \frac{2}{5} \qquad \mathrm{P}(A \cup B) = \frac{13}{20}\]

(a) Find \(\mathrm{P}(A \cap B)\). (2)

(b) Draw a Venn diagram to show the events \(A\), \(B\) and all the associated probabilities. (3)

Find

(d) \(\mathrm{P}(B \mid A)\) (2)

(e) \(\mathrm{P}(A' \cap B)\) (1)

Mark scheme (a)

Scheme	Marks
\(\mathrm{P}(A \cap B) = \mathrm{P}(A \mid B) \times \mathrm{P}(B)\) \(\mathrm{P}(A \cap B) = \dfrac{2}{5} \times \dfrac{1}{2} = \dfrac{1}{5}\)	M1 A1
	(2)

Notes

M1 for \(\dfrac{2}{5} \times \dfrac{1}{2}\) or a correct probability product expression and one correct prob. Ans only 2/2

Venn diagram: two intersecting circles A and B in a box; A only 3/20, intersection 1/5, B only 3/10, outside 7/20 — Mark scheme (b)

Scheme	Marks
2 intersecting circles and ‘\(\mathrm{P}(A \cap B)\)’	B1ft
\(\dfrac{3}{20}\) and \(\dfrac{3}{10}\)	B1
Box and \(\dfrac{7}{20}\)	B1
	(3)

Mark scheme (c)

Scheme	Marks
\(\left[\mathrm{P}(A) = \dfrac{3}{20} + \dfrac{1}{5}\right] = \dfrac{7}{20}\) or 0.35	B1ft
	(1)

Notes

B1ft for 0.35 (o.e.) if no Venn diagram or correct follow through from their diagram
or allow 0.35 (or correct ft) from correct working e.g. 0.65 – 0.5 + (a)
B0 for 0.35 if their diagram does not give 0.35 unless it comes from correct work
Don’t insist on P(\(A\)) =... but do not award for \(\mathrm{P}(A' \cap B') = \frac{7}{20}\)

Mark scheme (d)

Scheme	Marks
\(\mathrm{P}(B \mid A) = \dfrac{\mathrm{P}(A \cap B)}{\mathrm{P}(A)} = \dfrac{\frac{1}{5}}{\frac{7}{20}}\)	M1
\(= \dfrac{4}{7}\)	A1 cao
	(2)

Notes

M1 for \(\dfrac{\text{their (a)}}{\text{their (c)}}\) or a correct ratio of probabilities from their diagram
NB incorrect use of \(\mathrm{P}(A' \cap B') = \frac{7}{20}\) scores M0 and num \(\geqslant\) denom scores M0

A1 for \(\frac{4}{7}\) only