S1 January 2011 Q7

EdexcelOld spec17 marksProbability

7. The bag \(P\) contains 6 balls of which 3 are red and 3 are yellow.
The bag \(Q\) contains 7 balls of which 4 are red and 3 are yellow.
A ball is drawn at random from bag \(P\) and placed in bag \(Q\). A second ball is drawn at random from bag \(P\) and placed in bag \(Q\).
A third ball is then drawn at random from the 9 balls in bag \(Q\).

The event \(A\) occurs when the 2 balls drawn from bag \(P\) are of the same colour.
The event \(B\) occurs when the ball drawn from bag \(Q\) is red.

(a) Complete the tree diagram shown below.

Tree diagram with three stages: first ball Red 1/2 or Yellow 1/2; after Red, second ball Red 2/5 or Yellow 3/5; after Yellow, second ball Red and Yellow with blank probabilities; third ball Red or Yellow on each branch, with 5/9 given for Red after Red-Yellow and for Red after Yellow-Red, all other probabilities blank

(4)

(b) Find \(\mathrm{P}(A)\) (3)

(d) Show that \(\mathrm{P}(A \cap B) = \dfrac{2}{9}\) (2)

(e) Hence find \(\mathrm{P}(A \cup B)\) (2)

(f) Given that all three balls drawn are the same colour, find the probability that they are all red. (3)

Scheme	Marks

both \(\dfrac{2}{3}, \dfrac{1}{3}\)	B1
\(\dfrac{4}{9}\)	B1
both \(\dfrac{3}{5}, \dfrac{2}{5}\)	B1
all three of \(\dfrac{4}{9}, \dfrac{4}{9}, \dfrac{5}{9}\)	B1
	(4)

Mark scheme (b)

Scheme	Marks
\(\mathrm{P}(A) = \mathrm{P}(RR) + \mathrm{P}(YY) = \dfrac{1}{2} \times \dfrac{2}{5} + \dfrac{1}{2} \times \text{"}\dfrac{2}{5}\text{"} = \dfrac{2}{5}\) B1 for \(\dfrac{1}{2} \times \dfrac{2}{5}\) (oe) seen at least once	B1 M1 A1
	(3)

Notes

M1 for both cases, and +, attempted, ft their values from tree diagram. May be 4 cases of 3 balls.

Mark scheme (c)

Scheme	Marks
\(\mathrm{P}(B) = \mathrm{P}(RRR) + \mathrm{P}(RYR) + \mathrm{P}(YRR) + \mathrm{P}(YYR)\) M1 for at least 1 case of 3 balls identified. (Implied by 2nd M1)	M1
\(\left(\dfrac{1}{2} \times \dfrac{2}{5} \times \text{"}\dfrac{2}{3}\text{"}\right) + \left(\dfrac{1}{2} \times \dfrac{3}{5} \times \dfrac{5}{9}\right) + \left(\dfrac{1}{2} \times \text{"}\dfrac{3}{5}\text{"} \times \dfrac{5}{9}\right) + \left(\dfrac{1}{2} \times \text{"}\dfrac{2}{5}\text{"} \times \text{"}\dfrac{4}{9}\text{"}\right) = \dfrac{5}{9}\) (*)	M1,A1cso
	(3)

Notes

2nd M1 for all 4 correct expressions, ft their values from tree diagram. A1 is cso

Mark scheme (d)

Scheme	Marks
\(\mathrm{P}(A \cap B) = \mathrm{P}(RRR) + \mathrm{P}(YYR)\) M1 for identifying both cases and + probs. may be implied by correct expressions	M1
\(= \left(\dfrac{1}{2} \times \dfrac{2}{5} \times \dfrac{2}{3}\right) + \left(\dfrac{1}{2} \times \dfrac{2}{5} \times \dfrac{4}{9}\right) = \dfrac{2}{9}\) (*)	A1cso
	(2)

Mark scheme (e)

Scheme	Marks
\(\mathrm{P}(A \cup B) = \mathrm{P}(A) + \mathrm{P}(B) - \mathrm{P}(A \cap B)\) Must have some attempt to use	M1
\(= \text{"}\dfrac{2}{5}\text{"} + \dfrac{5}{9} - \dfrac{2}{9} = \dfrac{11}{15}\)	A1cao
	(2)

Notes

M1 for clear attempt to use the correct formula, must have some correct substitution. ft their (b)

Mark scheme (f)

Scheme	Marks
\(\dfrac{\mathrm{P}(RRR)}{\mathrm{P}(RRR) + \mathrm{P}(YYY)} = \dfrac{\frac{1}{2} \times \frac{2}{5} \times \frac{2}{3}}{\left(\frac{1}{2} \times \frac{2}{5} \times \frac{2}{3}\right) + \left(\frac{1}{2} \times \frac{2}{5} \times \frac{5}{9}\right)} = \dfrac{6}{11}\) Probabilities must come from the product of 3 probs. from their tree diagram.	M1 A1ft A1 cao
	(3)
	(17 marks)

Notes

M1 for identifying the correct probabilities and forming appropriate fraction of probs.
1st A1ft for a correct expression using probabilities from their tree

Accept exact decimal equivalents. Correct answer only is full marks except in (c) and (d)