| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics B AS (Further Statistics B AS) |
| Session | Specimen |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Probability |
| Type | Simulation design and interpretation |
| Difficulty | Easy -1.2 This is a straightforward simulation question requiring basic spreadsheet functions (RANDBETWEEN, MAX, MIN) and simple probability estimation by counting favorable outcomes. The conceptual demand is low—students just need to understand how to set up formulas and interpret simulation results. No complex probability theory or multi-step reasoning is required. |
| Spec | 2.03a Mutually exclusive and independent events |
| C5 | \(\times \vee f _ { x }\) | =RANDBETWEEN(1,6) | ||||||||
| A | B | C | D | E | F | G | H | J | ||
| 1 | dice 1 | dice 2 | dice 3 | dice 4 | dice 5 | dice 6 | High score | Low score | Difference | |
| 2 | game 1 | 2 | 2 | 4 | 2 | 3 | 3 | 4 | 2 | 2 |
| 3 | game 2 | 2 | 6 | 3 | 2 | 1 | 2 | 6 | 1 | 5 |
| 4 | game 3 | 3 | 1 | 5 | 3 | 4 | 6 | 6 | 1 | 5 |
| 5 | game 4 | 6 | 5 | 2 | 5 | 6 | 3 | 6 | 2 | 4 |
| 6 | game 5 | 6 | 3 | 3 | 5 | 3 | 2 | 6 | 2 | 4 |
| 7 | game 6 | 5 | 6 | 3 | 5 | 1 | 4 | 6 | 1 | 5 |
| 8 | game 7 | 2 | 3 | 1 | 2 | 6 | 4 | 6 | 1 | 5 |
| 9 | game 8 | 6 | 6 | 6 | 6 | 1 | 5 | 6 | 1 | 5 |
| 10 | game 9 | 3 | 6 | 2 | 5 | 4 | 1 | 6 | 1 | 5 |
| 11 | game 10 | 5 | 1 | 1 | 4 | 6 | 1 | 6 | 1 | 5 |
| 12 | game 11 | 2 | 5 | 6 | 1 | 6 | 5 | 6 | 1 | 5 |
| 13 | game 12 | 2 | 5 | 6 | 6 | 6 | 6 | 6 | 2 | 4 |
| 14 | game 13 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 |
| 15 | game 14 | 1 | 6 | 6 | 6 | 3 | 5 | 6 | 1 | 5 |
| 16 | game 15 | 2 | 2 | 3 | 3 | 5 | 1 | 5 | 1 | 4 |
| 17 | game 16 | 1 | 2 | 3 | 4 | 3 | 3 | 4 | 1 | 3 |
| 18 | game 17 | 5 | 2 | 4 | 2 | 1 | 6 | 6 | 1 | 5 |
| 19 | game 18 | 6 | 1 | 5 | 2 | 1 | 5 | 6 | 1 | 5 |
| 20 | game 19 | 1 | 3 | 5 | 1 | 3 | 5 | 5 | 1 | 4 |
| 21 | game 20 | 5 | 4 | 3 | 2 | 5 | 1 | |||
| Answer | Marks | Guidance |
|---|---|---|
| 1 | (i) | (A) |
| [1] | 1.1 | |
| 1 | (i) | (B) |
| Answer | Marks |
|---|---|
| =0.1 | M1 |
| Answer | Marks |
|---|---|
| [2] | 3.4 |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | (ii) | Simulate more goes at the game |
| [1] | 3.5c | N |
| 1 | (iii) | Advantage is that the probabilities are difficult to |
| Answer | Marks |
|---|---|
| Disadvantage is that it does not give an exact answer | E1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 3.5b | E |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | (iv) | Expected profit = 50 × £1 – 50 × 0.1 × £5 |
| =£25 | M1 |
| Answer | Marks |
|---|---|
| [3] | I |
| Answer | Marks |
|---|---|
| 1.1 | For 50 × £1 |
Question 1:
1 | (i) | (A) | 5, 1, 4 | B1
[1] | 1.1
1 | (i) | (B) | 2
Estimate of P(Less than 3) =
20
=0.1 | M1
A1
[2] | 3.4
1.1
1 | (ii) | Simulate more goes at the game | E1
[1] | 3.5c | N
1 | (iii) | Advantage is that the probabilities are difficult to
calculate
Disadvantage is that it does not give an exact answer | E1
E1
[2] | 1.1
3.5b | E
M
1 | (iv) | Expected profit = 50 × £1 – 50 × 0.1 × £5
=£25 | M1
C
M1
A1
[3] | I
3.4
1.1a
1.1 | For 50 × £1
for 50 × 0.1 × £51 Abby runs a stall at a charity event. Visitors to the stall pay to play a game in which six fair dice are rolled. If the difference between the highest and lowest scores is less than 3 then the player wins $\pounds 5$. Otherwise the player wins nothing.
Abby designs the spreadsheet shown in Fig. 1 to estimate the probability of a player winning, by simulating 20 goes at the game. Cell C5, highlighted, shows that the 2nd dice in simulated game 4 scores 5 . Cells H5 and I5 show the highest and lowest scores, respectively, in game 4, and cell J5 gives the difference between them.
\begin{table}[h]
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline
\multicolumn{3}{|l|}{C5} & \multicolumn{2}{|c|}{$\times \vee f _ { x }$} & \multicolumn{6}{|c|}{=RANDBETWEEN(1,6)} \\
\hline
& A & B & C & D & E & F & G & H & & J \\
\hline
1 & & dice 1 & dice 2 & dice 3 & dice 4 & dice 5 & dice 6 & High score & Low score & Difference \\
\hline
2 & game 1 & 2 & 2 & 4 & 2 & 3 & 3 & 4 & 2 & 2 \\
\hline
3 & game 2 & 2 & 6 & 3 & 2 & 1 & 2 & 6 & 1 & 5 \\
\hline
4 & game 3 & 3 & 1 & 5 & 3 & 4 & 6 & 6 & 1 & 5 \\
\hline
5 & game 4 & 6 & 5 & 2 & 5 & 6 & 3 & 6 & 2 & 4 \\
\hline
6 & game 5 & 6 & 3 & 3 & 5 & 3 & 2 & 6 & 2 & 4 \\
\hline
7 & game 6 & 5 & 6 & 3 & 5 & 1 & 4 & 6 & 1 & 5 \\
\hline
8 & game 7 & 2 & 3 & 1 & 2 & 6 & 4 & 6 & 1 & 5 \\
\hline
9 & game 8 & 6 & 6 & 6 & 6 & 1 & 5 & 6 & 1 & 5 \\
\hline
10 & game 9 & 3 & 6 & 2 & 5 & 4 & 1 & 6 & 1 & 5 \\
\hline
11 & game 10 & 5 & 1 & 1 & 4 & 6 & 1 & 6 & 1 & 5 \\
\hline
12 & game 11 & 2 & 5 & 6 & 1 & 6 & 5 & 6 & 1 & 5 \\
\hline
13 & game 12 & 2 & 5 & 6 & 6 & 6 & 6 & 6 & 2 & 4 \\
\hline
14 & game 13 & 2 & 2 & 2 & 2 & 4 & 4 & 4 & 2 & 2 \\
\hline
15 & game 14 & 1 & 6 & 6 & 6 & 3 & 5 & 6 & 1 & 5 \\
\hline
16 & game 15 & 2 & 2 & 3 & 3 & 5 & 1 & 5 & 1 & 4 \\
\hline
17 & game 16 & 1 & 2 & 3 & 4 & 3 & 3 & 4 & 1 & 3 \\
\hline
18 & game 17 & 5 & 2 & 4 & 2 & 1 & 6 & 6 & 1 & 5 \\
\hline
19 & game 18 & 6 & 1 & 5 & 2 & 1 & 5 & 6 & 1 & 5 \\
\hline
20 & game 19 & 1 & 3 & 5 & 1 & 3 & 5 & 5 & 1 & 4 \\
\hline
21 & game 20 & 5 & 4 & 3 & 2 & 5 & 1 & & & \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{table}
\begin{enumerate}[label=(\roman*)]
\item (A) Write down the numbers in columns H , I and J for game 20 .\\
(B) Use the spreadsheet to estimate the probability of a player winning a game.
\item State how the estimate of probability in (i) (B) could be improved.
\item Give one advantage and one disadvantage of using this simulation technique compared with working out the theoretical probability.
All profit made by the stall is given to charity. Abby has to decide how much to charge players to play.
\item If Abby charges $\pounds 1$ per game, estimate the total profit when 50 players each play the game once.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics B AS Q1 [9]}}