A2 October 2020 Q3

EdexcelCurrent spec9 marksGeometric & Negative Binomial

3. Suzanne and Jon are playing a game.

They put 4 red counters and 1 blue counter in a bag.

Suzanne reaches into the bag and selects one of the counters at random. If the counter she selects is blue, she wins the game. Otherwise she puts it back in the bag and Jon selects one at random. If the counter he selects is blue, he wins the game. Otherwise he puts it back in the bag and they repeat this process until one of them selects the blue counter.

(a) Find the probability that Suzanne selects the blue counter on her 4th selection. (2)

(b) Find the probability that the blue counter is first selected on or after Jon’s third selection. (2)

(d) Find the probability that Suzanne wins the game. (3)

Mark scheme (a)

Scheme	Marks	AO
[\(X \sim \mathrm{Geo}(0.2)\) Suzanne’s 4^th selection is the 7^th selection overall] \(\mathrm{P}(X = 7) = (0.8)^6(0.2)\) or \((0.64)^3(0.2)\)	M1	3.3
\(= 0.05242\ldots\) awrt 0.0524	A1	1.1b
	(2)

Notes

M1: Selecting geometric distribution with \(p = 0.2\) and attempting required probability.
Allow \((0.8)^n(0.2)\) to imply M1 with \(n = 6\) or \(n = 3\)

A1: awrt 0.0524 Allow exact fraction \(\dfrac{4096}{78125}\)

Mark scheme (b)

Scheme	Marks	AO
\(\mathrm{P}(X \geqslant 6)\ [= (1 - 0.2)^5]\)	M1	1.1b
\(= 0.32768\) awrt 0.328	A1	1.1b
	(2)

Notes

M1: \(\mathrm{P}(X \geqslant 6)\) may be implied by \((1 - p)^5\) or \(1 - (p + pq + pq^2 + pq^3 + pq^4)\)

A1: awrt 0.328 Allow exact fraction \(\dfrac{1024}{3125}\)

Mark scheme (c)

Scheme	Marks	AO
Mean = 5	B1	1.1b
Standard deviation \(\left[= \sqrt{\dfrac{1 - 0.2}{0.2^2}}\right] = \sqrt{20}\) awrt 4.47	B1	1.1b
	(2)

Notes

B1: Mean = 5

B1: Standard deviation = \(\sqrt{20}\) o.e. or awrt 4.47

Mark scheme (d)

Scheme	Marks	AO
\(\mathrm{P}(\text{Suzanne wins}) = 0.2 + (0.8)^2(0.2) + (0.8)^4(0.2) + \ldots\)	M1	3.1b
Infinite geometric series \(= \dfrac{0.2}{1 - 0.8^2}\) (oe)	M1	2.1
\(= \dfrac{5}{9}\)	A1	1.1b
	(3)
	(9 marks)

Notes

M1: Determining the probability that Suzanne wins with at least three terms seen (may be implied by 2^nd M1)

M1: Recognising need to sum terms of an infinite geometric series with correct \(r = 0.8^2\) (with numerator less than denominator)

A1: \(\dfrac{5}{9}\) (allow awrt 0.556)