S1 June 2011 Q6

EdexcelOld spec9 marksProbability

6. Jake and Kamil are sometimes late for school.
The events \(J\) and \(K\) are defined as follows

\(J\) = the event that Jake is late for school
\(K\) = the event that Kamil is late for school

\[\mathrm{P}(J) = 0.25, \quad \mathrm{P}(J \cap K) = 0.15 \quad \text{and} \quad \mathrm{P}(J' \cap K') = 0.7\]

On a randomly selected day, find the probability that

(a) at least one of Jake or Kamil are late for school, (1)

(b) Kamil is late for school. (2)

Given that Jake is late for school,

The teacher suspects that Jake being late for school and Kamil being late for school are linked in some way.

(d) Determine whether or not \(J\) and \(K\) are statistically independent. (2)

(e) Comment on the teacher’s suspicion in the light of your calculation in (d). (1)

Scheme	Marks
\(\mathrm{P}(J \cup K) = 1 - 0.7\) or \(0.1 + 0.15 + 0.05 =\) 0.3	B1
	(1)

Venn diagram: J only 0.10, J and K 0.15, K only 0.05, outside 0.7 — Mark scheme (b)

Scheme	Marks
P(\(K\)) = 0.05 + 0.15 or “0.3” – 0.25 + 0.15 or “0.3” = 0.25 +P(\(K\)) – 0.15	M1
May be seen on Venn diagram = 0.2	A1
	(2)

Mark scheme (c)

Scheme	Marks
\(\left[\mathrm{P}(K \mid J)\right] = \dfrac{\mathrm{P}(K \cap J)}{\mathrm{P}(J)}\)	M1
\(= \dfrac{0.15}{0.25}\)	A1
\(= \underline{\dfrac{3}{5}}\) or 0.6	A1
	(3)

Notes

In parts (c) and (d) they must have defined \(A\) and \(B\)

M1 for a correct expression (including ratio) in symbols.
1st A1 for a correct ratio of probabilities (if this is seen the M1 is awarded by implication)
Must be in (c). Condone no LHS but wrong LHS (e.g. P(\(K\)) or P(\(J|K\))) is M0A0
2nd A1 for correct answer as printed only. Correct answer only 3/3

Mark scheme (d)

Scheme	Marks
\(\mathrm{P}(J) \times \mathrm{P}(K) = 0.25 \times 0.2\ (= 0.05),\ \ \mathrm{P}(J \cap K) = 0.15\) or \(\mathrm{P}(K \mid J) = 0.6,\ \mathrm{P}(K) = 0.2\) or may see P(\(J\|K\)) = 0.75 and P(\(J\)) = 0.25	M1
not equal therefore not independent	A1ft
	(2)

Notes

Mark (d) and (e) together

M1 for a correct comparison of known probabilities for an independence test - ft their values. E.g. \(\mathrm{P}(J) \times \mathrm{P}(K)\) with \(\mathrm{P}(J \cap K)\) or P(\(K|J\)) with P(\(K\)) [Must have expressions]
The values of these probabilities should be given unless they are in the question or stated elsewhere.
A1ft for correct calculations and correct comment for their probabilities

Mark scheme (e)

Scheme	Marks
Not independent so confirms the teacher’s suspicion or they are linked (This requires a statement about independence in (d) or in (e))	B1ft
	(1)
	(9 marks)

Notes

B1ft ft their conclusion on independence so not independent confirms teacher…independent contradicts teacher.

Methods leading to negative probabilities should score M0