| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2013 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Construct probability distribution from scenario |
| Difficulty | Standard +0.3 This is a straightforward S1 question testing basic probability distribution concepts. Parts (a)-(e) involve routine calculations: writing distributions, identifying discrete uniform distribution, and computing E(X) and Var(X) using standard formulas. Part (f) requires conditional probability reasoning but is systematic—compare P(B>2), P(R>2), P(B>5), P(R>5), then apply the law of total probability. All techniques are standard S1 content with no novel insight required. |
| Spec | 2.04a Discrete probability distributions5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables5.02e Discrete uniform distribution |
| \(r\) | 2 | 4 | 6 |
| \(\mathrm { P } ( R = r )\) | \(\frac { 2 } { 3 }\) | \(\frac { 1 } { 6 }\) | \(\frac { 1 } { 6 }\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(b\) values: \(1, 3, 5\) | B1 | Correctly identifying \(b\) as 1, 3, 5 (or 1,1,3,3,5,5) |
| \(P(B=b)=\frac{1}{3}\) each (or all \(\frac{1}{6}\)) | B1 | Any correct probability distribution or function is 2/2 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Discrete Uniform {distribution} | B1 | Both words required |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(E(B)=3\) (by symmetry) | B1 | Accept \(E(X)=3\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(E(R)=2\times\frac{2}{3}+4\times\frac{1}{6}+6\times\frac{1}{6}\) | M1 | At least 2 correct products; if divide by \(n(\neq 1)\) then M0 |
| \(=\mathbf{3}\) | A1 | Correct answer only scores both marks |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(E(R^2)=2^2\times\frac{2}{3}+4^2\times\frac{1}{6}+6^2\times\frac{1}{6}\) \(\left[=\frac{34}{3}\right]\) | M1 | At least 2 correct products; may be implied by \(\frac{34}{3}\) or \(11.3\) or better |
| \(\text{Var}(R)=\frac{34}{3}-3^2=\frac{7}{3}\) (any exact equivalent; NB 2.33 is A0) | dM1, A1 | Dependent on 1st M1; clear attempt at \(E(R^2)-[E(R)]^2\) with values used; \(\text{Var}(R)=E(R^2)-\frac{34}{3}\text{"-"3"}\) is M1M0A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Coin lands on 2 → choose blue die; coin lands on 5 → choose red die | B2/1/0 | Both correct B1B1; one correct B1B0; do not use B0B1 |
| \(P(\text{Avisha wins})=\frac{1}{2}\times\left(\frac{1}{3}+\frac{1}{3}\right)+\frac{1}{2}\times\frac{1}{6}\) | M1 | Evaluating correct probabilities: only \(\frac{1}{3},\frac{1}{12}\) seen or if incorrect choice made: RR \((\frac{1}{4})\), BB \((\frac{1}{3})\), RB \((\frac{1}{6})\) |
| \(=\frac{5}{12}\) (allow awrt \(0.417\)) | A1 | \(\frac{5}{12}\) as answer scores M1A1; need choices of die stated for B marks |
# Question 6:
## Part (a)
| Working | Mark | Guidance |
|---------|------|---------|
| $b$ values: $1, 3, 5$ | B1 | Correctly identifying $b$ as 1, 3, 5 (or 1,1,3,3,5,5) |
| $P(B=b)=\frac{1}{3}$ each (or all $\frac{1}{6}$) | B1 | Any correct probability distribution or function is 2/2 |
## Part (b)
| Working | Mark | Guidance |
|---------|------|---------|
| Discrete Uniform {distribution} | B1 | Both words required |
## Part (c)
| Working | Mark | Guidance |
|---------|------|---------|
| $E(B)=3$ (by symmetry) | B1 | Accept $E(X)=3$ |
## Part (d)
| Working | Mark | Guidance |
|---------|------|---------|
| $E(R)=2\times\frac{2}{3}+4\times\frac{1}{6}+6\times\frac{1}{6}$ | M1 | At least 2 correct products; if divide by $n(\neq 1)$ then M0 |
| $=\mathbf{3}$ | A1 | Correct answer only scores both marks |
## Part (e)
| Working | Mark | Guidance |
|---------|------|---------|
| $E(R^2)=2^2\times\frac{2}{3}+4^2\times\frac{1}{6}+6^2\times\frac{1}{6}$ $\left[=\frac{34}{3}\right]$ | M1 | At least 2 correct products; may be implied by $\frac{34}{3}$ or $11.3$ or better |
| $\text{Var}(R)=\frac{34}{3}-3^2=\frac{7}{3}$ (any exact equivalent; NB 2.33 is A0) | dM1, A1 | Dependent on 1st M1; clear attempt at $E(R^2)-[E(R)]^2$ with values used; $\text{Var}(R)=E(R^2)-\frac{34}{3}\text{"-"3"}$ is M1M0A0 |
## Part (f)
| Working | Mark | Guidance |
|---------|------|---------|
| Coin lands on 2 → choose **blue** die; coin lands on 5 → choose **red** die | B2/1/0 | Both correct B1B1; one correct B1B0; do not use B0B1 |
| $P(\text{Avisha wins})=\frac{1}{2}\times\left(\frac{1}{3}+\frac{1}{3}\right)+\frac{1}{2}\times\frac{1}{6}$ | M1 | Evaluating correct probabilities: only $\frac{1}{3},\frac{1}{12}$ seen or if incorrect choice made: RR $(\frac{1}{4})$, BB $(\frac{1}{3})$, RB $(\frac{1}{6})$ |
| $=\frac{5}{12}$ (allow awrt $0.417$) | A1 | $\frac{5}{12}$ as answer scores M1A1; need choices of die stated for B marks |6. A fair blue die has faces numbered $1,1,3,3,5$ and 5 . The random variable $B$ represents the score when the blue die is rolled.
\begin{enumerate}[label=(\alph*)]
\item Write down the probability distribution for $B$.
\item State the name of this probability distribution.
\item Write down the value of $\mathrm { E } ( B )$.
A second die is red and the random variable $R$ represents the score when the red die is rolled.
The probability distribution of $R$ is
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
$r$ & 2 & 4 & 6 \\
\hline
$\mathrm { P } ( R = r )$ & $\frac { 2 } { 3 }$ & $\frac { 1 } { 6 }$ & $\frac { 1 } { 6 }$ \\
\hline
\end{tabular}
\end{center}
\item Find $\mathrm { E } ( R )$.
\item Find $\operatorname { Var } ( R )$.
Tom invites Avisha to play a game with these dice.\\
Tom spins a fair coin with one side labelled 2 and the other side labelled 5 . When Avisha sees the number showing on the coin she then chooses one of the dice and rolls it. If the number showing on the die is greater than the number showing on the coin, Avisha wins, otherwise Tom wins.
Avisha chooses the die which gives her the best chance of winning each time Tom spins the coin.
\item Find the probability that Avisha wins the game, stating clearly which die she should use in each case.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2013 Q6 [13]}}