October 2021 Paper 2 Q15

OCR MEICurrent spec11 marksDRVs

(a) Show that \(\displaystyle\sum_{r=1}^{\infty} 0.99^{r-1} \times 0.01 = 1\). [3]

Kofi is a very good table tennis player. Layla is determined to beat him.

Every week they play one match of table tennis against each other. They will stop playing when Layla wins the match for the first time.

\(X\) is the discrete random variable “the number of matches they play in total”.

Kofi models the situation using the probability function

\(\mathrm{P}(X = r) = 0.99^{r-1} \times 0.01 \qquad r = 1, 2, 3, 4, \ldots\)

Kofi states that he is 95% certain that Layla will not beat him within 6 years.

(b) Determine whether Kofi’s statement is consistent with his model. [3]

In between matches, Layla practises, but Kofi does not.

Layla models the situation using the probability function

\(\mathrm{P}(X = r) = kr^2 \qquad r = 1, 2, 3, 4, 5, 6, 7, 8.\)

(d) Explain how Layla’s model takes into account the fact that she practises between matches, but Kofi’s does not. [1]

Layla states that she is 95% certain that she will beat Kofi within the first 6 matches.

(e) Determine whether Layla’s statement is consistent with her model. [3]

Mark scheme (a)

Scheme	Marks	AO
\(a = 0.01\) and \(r = 0.99\)	B1	2.1
\(a\) and \(r\) substituted in \(\frac{a}{1-r}\) oe seen	M1	1.1
\(\dfrac{0.01}{1-0.99}\) oe \(= 1\)	A1	2.4
	[3]

M1: \(a\) or \(r\) must be correct

Mark scheme (b)

Scheme	Marks	AO
\((n =)\ 312\)	M1	3.1a
\(\dfrac{0.01(1 - 0.99^n)}{1 - 0.99}\) or \(\sum_1^n 0.99^{n-1} \times 0.01\) evaluated	M1	1.1
awrt 0.9565 or 0.9570 > 0.95 so model predicts Layla will beat him within 6 years oe	A1	3.4
	[3]

M1: allow 313

M1: \(n = 312\) or 313; condone \(n = 6\)

Scheme	Marks
\(\dfrac{0.01(1 - 0.99^n)}{1 - 0.99} = 0.95\)	M1
\(0.99^n = 0.05\)	M1
\(n = 298.1\) weeks \(< 312\) (or 313) so model predicts Layla will beat him within 6 years oe	A1

M1: may use cdf from geometric distribution;
allow \(>\) or \(\geqslant\) instead of \(=\)

M1: attempt to simplify as far as “\(0.99^n =\)”

Mark scheme (c)

Scheme	Marks	AO
Layla thinks she will improve (by practising), so she should become increasingly likely to beat Kofi oe or Layla thinks she is more likely to beat Kofi because he doesn’t practise (but Layla does) oe	B1	3.5a
	[1]

Scheme	Marks	AO
probability of Layla winning increases as \(r\) increases oe	B1	2.4
	[1]

Mark scheme (e)

Scheme	Marks	AO
\(k(1 + 4 + 9 + 16 + 25 + 36 + 49 + 64) = 1\) soi	M1	3.3
\(k = \frac{1}{204}\ (= 0.00490\ldots)\)	A1	1.1
\(\mathrm{P}(X \leqslant 6) = \frac{91}{204}\ (0.44607\ldots) < 0.95\) so Layla’s statement not consistent with her model oe	A1	3.5a
	[3]