| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics Major (Further Statistics Major) |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Sequential trials until success |
| Difficulty | Standard +0.3 This is a straightforward application of geometric/waiting time probability with a small state space. Part (i) is a simple calculation (1 × 2/3 × 1/3 × 3! arrangements = 2/9). Part (ii) requires systematic enumeration of cases but with only 3 types and max 6 trials, the calculations are routine. Part (iii) is standard expectation and variance from a discrete distribution table. While it requires careful bookkeeping across multiple cases, it involves no novel insight and is typical of Further Statistics questions at this level. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| \(r\) | 3 | 4 | 5 | 6 |
| \(\mathrm { P } ( X = r )\) | \(\frac { 2 } { 9 }\) | \(\frac { 14 } { 81 }\) |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | (i) | 2 |
| 9 | B1 | |
| [1] | 1.1 | |
| 1 | (ii) | r 3 4 5 6 |
| Answer | Marks | Guidance |
|---|---|---|
| 9 81 | B1 | |
| [1] | 1.1 | N |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | (iii) | DR |
| Answer | Marks |
|---|---|
| =1.413 | M1 |
| Answer | Marks |
|---|---|
| [5] | I1.2 |
| Answer | Marks |
|---|---|
| 1.1 | M |
| Answer | Marks | Guidance |
|---|---|---|
| r | 3 | 4 |
| P(X = r) | 2 | |
| 9 | 31 |
Question 1:
1 | (i) | 2
9 | B1
[1] | 1.1
1 | (ii) | r 3 4 5 6
2 31
P(X = r)
9 81 | B1
[1] | 1.1 | N
E
1 | (iii) | DR
2 2 14 31
E(X)(cid:32)3(cid:117) (cid:14)4(cid:117) (cid:14)5(cid:117) (cid:14)6(cid:117)
9 9 81 81
382
(cid:32) (cid:32)4.716
81
2 2 14 31
E(X2)(cid:32)9(cid:117) (cid:14)16(cid:117) (cid:14)25(cid:117) (cid:14)36(cid:117)
9 9 81 81
P
1916
(cid:32) (cid:32)23.564
81
S
2
1916 (cid:167)382(cid:183)
Var(X)(cid:32) (cid:16)(cid:168) (cid:184)
81 (cid:169) 81 (cid:185)
=1.413 | M1
A1
E
M1
M1
A1
[5] | I1.2
C
1.1
1.1
1.1
1.1 | M
For Σrp (at least 3 terms correct)
FT their part (ii)
cao
For Σr2p (at least 3 terms
correct)
M1dep for – their E( X )²
A1 FT their E(X) provided
Var( X ) > 0
r | 3 | 4 | 5 | 6
P(X = r) | 2
9 | 31
811 In a promotion for a new type of cereal, a toy dinosaur is included in each pack. There are three different types of dinosaur to collect. They are distributed, with equal probability, randomly and independently in the packs. Sam is trying to collect all three of the dinosaurs.\\
(i) Find the probability that Sam has to open only 3 packs in order to collect all three dinosaurs.
Sam continues to open packs until she has collected all three dinosaurs, but once she has opened 6 packs she gives up even if she has not found all three. The random variable $X$ represents the number of packs which Sam opens.\\
(ii) Complete the table below, using the copy in the Printed Answer Booklet, to show the probability distribution of $X$.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | }
\hline
$r$ & 3 & 4 & 5 & 6 \\
\hline
$\mathrm { P } ( X = r )$ & & $\frac { 2 } { 9 }$ & $\frac { 14 } { 81 }$ & \\
\hline
\end{tabular}
\end{center}
\section*{(iii) In this question you must show detailed reasoning.}
Find
\begin{itemize}
\item $\mathrm { E } ( X )$ and
\item $\operatorname { Var } ( X )$.
\end{itemize}
\hfill \mbox{\textit{OCR MEI Further Statistics Major Q1 [7]}}