| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics B AS (Further Statistics B AS) |
| Year | 2019 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Construct probability distribution from scenario |
| Difficulty | Moderate -0.8 This is a straightforward question requiring basic understanding of probability distributions from a given scenario. Parts (a)-(c) involve simple counting from a spreadsheet simulation with no calculation complexity. The most demanding aspect is constructing the distributions, but for Further Maths Stats students, identifying that L follows a standard two-dice distribution and C follows a binomial distribution (scaled by 3) is routine. This is easier than average even for Further Maths. |
| Spec | 2.03a Mutually exclusive and independent events2.04b Binomial distribution: as model B(n,p) |
| A | B | C | D | E | F | G | H | |
| 1 | First dice | Second dice | Total (Leila's score) \(\boldsymbol { L }\) | Coin 1 | Coin 2 | Coin 3 | Coin 4 | Caleb's score \(\boldsymbol { C }\) |
| 2 | 1 | 2 | 3 | H | T | T | T | 3 |
| 3 | 6 | 1 | 7 | T | H | T | T | 3 |
| 4 | 2 | 6 | 8 | H | H | T | T | 6 |
| 5 | 2 | 5 | 7 | T | H | H | H | 9 |
| 6 | 1 | 5 | 6 | T | H | T | T | 3 |
| 7 | 5 | 2 | 7 | H | H | H | H | 12 |
| 8 | 1 | 1 | 2 | H | T | H | T | 6 |
| 9 | 2 | 6 | 8 | T | H | T | H | 6 |
| 10 | 6 | 2 | 8 | H | T | H | T | 6 |
| 11 | 1 | 3 | 4 | T | H | H | H | 9 |
| 12 | 6 | 1 | 7 | T | H | T | T | 3 |
| 13 | 3 | 1 | 4 | T | T | T | T | 0 |
| 14 | 3 | 6 | 9 | H | T | H | H | 9 |
| 15 | 2 | 3 | 5 | T | H | H | H | 9 |
| 16 | 2 | 5 | 7 | H | H | H | H | 12 |
| 17 | 1 | 5 | 6 | H | H | T | H | 9 |
| 18 | 5 | 6 | 11 | T | H | H | H | 9 |
| 19 | 4 | 2 | 6 | T | H | H | T | 6 |
| 20 | 6 | 5 | 11 | T | T | H | H | 6 |
| 21 | 1 | 1 | 2 | T | T | T | T | 0 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (a) | Because there is one head and the score is 3 × |
| number of heads. | E1 | |
| [1] | Watch for candidates who |
| Answer | Marks |
|---|---|
| (b) | Estimate of P(C > 6) = 8 = 0.4 |
| Answer | Marks |
|---|---|
| 20 | B1 |
| Answer | Marks |
|---|---|
| (c) | P(Leila loses) = 7 |
| Answer | Marks |
|---|---|
| = 0.35 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| (d) | Because the number of simulations is small. | E1 |
| Answer | Marks |
|---|---|
| (e) | Caleb did not win 50% of games as there were |
| Answer | Marks |
|---|---|
| Comparing the maximum values is not sensible. | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| [2] | For any two valid comments | Do not allow comments about |
Question 2:
2 | (a) | Because there is one head and the score is 3 ×
number of heads. | E1
[1] | Watch for candidates who
work out Leila’s score
(b) | Estimate of P(C > 6) = 8 = 0.4
20
Estimate of P(L > 6) = 11 = 0.55
20 | B1
B1
[2]
(c) | P(Leila loses) = 7
20
= 0.35 | M1
A1
[2]
(d) | Because the number of simulations is small. | E1
[1]
(e) | Caleb did not win 50% of games as there were
some draws, so may not be true.
A much larger sample would be needed to be
reasonably sure of the conclusion.
Although both can get a max score of 12, Leila’s
minimum is 2 as compared to Caleb’s 0, so may
not be true.
Leila’s expected score is 7 but Caleb’s is 6, so
may not be true.
Comparing the maximum values is not sensible. | E1
E1
[2] | For any two valid comments | Do not allow comments about
simulation not being able to
account for bias unless
mention ‘due to small
sample’ oe2 Leila and Caleb are playing a game, using fair six-sided dice and unbiased coins.
\begin{itemize}
\item Leila rolls two dice, and her score $L$ is the total of the scores on the two dice.
\item Caleb spins 4 coins and his score $C$ is three times the number of heads obtained.
\end{itemize}
The winner of a game is the player with the higher score. If the two scores are equal, the result of the game is a draw. The spreadsheet in Fig. 2 shows a simulation of 20 plays of the game.
\begin{table}[h]
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
& A & B & C & D & E & F & G & H \\
\hline
1 & First dice & Second dice & Total (Leila's score) $\boldsymbol { L }$ & Coin 1 & Coin 2 & Coin 3 & Coin 4 & Caleb's score $\boldsymbol { C }$ \\
\hline
2 & 1 & 2 & 3 & H & T & T & T & 3 \\
\hline
3 & 6 & 1 & 7 & T & H & T & T & 3 \\
\hline
4 & 2 & 6 & 8 & H & H & T & T & 6 \\
\hline
5 & 2 & 5 & 7 & T & H & H & H & 9 \\
\hline
6 & 1 & 5 & 6 & T & H & T & T & 3 \\
\hline
7 & 5 & 2 & 7 & H & H & H & H & 12 \\
\hline
8 & 1 & 1 & 2 & H & T & H & T & 6 \\
\hline
9 & 2 & 6 & 8 & T & H & T & H & 6 \\
\hline
10 & 6 & 2 & 8 & H & T & H & T & 6 \\
\hline
11 & 1 & 3 & 4 & T & H & H & H & 9 \\
\hline
12 & 6 & 1 & 7 & T & H & T & T & 3 \\
\hline
13 & 3 & 1 & 4 & T & T & T & T & 0 \\
\hline
14 & 3 & 6 & 9 & H & T & H & H & 9 \\
\hline
15 & 2 & 3 & 5 & T & H & H & H & 9 \\
\hline
16 & 2 & 5 & 7 & H & H & H & H & 12 \\
\hline
17 & 1 & 5 & 6 & H & H & T & H & 9 \\
\hline
18 & 5 & 6 & 11 & T & H & H & H & 9 \\
\hline
19 & 4 & 2 & 6 & T & H & H & T & 6 \\
\hline
20 & 6 & 5 & 11 & T & T & H & H & 6 \\
\hline
21 & 1 & 1 & 2 & T & T & T & T & 0 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{center}
\end{table}
\begin{enumerate}[label=(\alph*)]
\item Explain why the value of $C$ in row 2 is 3 .
\item Use the spreadsheet to estimate $\mathrm { P } ( C > 6 )$ and $\mathrm { P } ( L > 6 )$.
\item Use the spreadsheet to estimate the probability that Leila loses a randomly chosen game.
\item Explain why your answers to parts (b) and (c) may not be very close to the true values.
\item Leila claims that the game is fair (that Leila and Caleb each have an equal chance of winning) because both she and Caleb can get a maximum score of 12 and also in the simulation she won exactly $50 \%$ of the games.\\
Make 2 comments about Leila's claim.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics B AS 2019 Q2 [8]}}