| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2022 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Calculate Var(X) from table |
| Difficulty | Moderate -0.8 This is a standard S1 question testing routine probability distribution calculations: verifying E(X²), finding standard deviation using Var(X) = E(X²) - [E(X)]², completing a probability table, and basic probability calculations. All parts follow textbook procedures with no novel problem-solving required, making it easier than average for A-level. |
| Spec | 2.04a Discrete probability distributions5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| \(r\) | 2 | 3 | 4 | 5 | 6 |
| \(\mathrm { P } ( R = r )\) | 0.25 | 0.3 | 0.15 | 0.1 | 0.2 |
| \(y\) | 2 | 3 | 4 | 5 | 6 |
| \(\mathrm { P } ( Y = y )\) | 0.1 | 0.2 | 0.1 | \(a\) | \(b\) |
| \(\mathrm {~F} ( y )\) | 0.1 | 0.3 | 0.4 | \(c\) | \(d\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(E(R^2) = 2^2 \times 0.25 + 3^2 \times 0.3 + 4^2 \times 0.15 + 5^2 \times 0.1 + 6^2 \times 0.2 (= 15.8^*)\) | B1cso* | Correct calculation with all products seen (allow \(1 + 2.7 + 2.4 + 2.5 + 7.2\)). Figures may be seen in table before part (a). Condone missing addition signs if products seen in table |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\text{sd}(R) = \sqrt{15.8 - 3.7^2}\) | M1 | Use of formula including square root |
| \(= \sqrt{2.11}\) | ||
| Standard deviation \(= 1.4525\ldots\) awrt 1.45 | A1 | Correct answer with no working scores M1A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(d = 1\) | B1 | For 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(0.1 + 0.2 + 0.1 + a + b = 1\) oe | M1 | Allow equivalents e.g. \(a + b = 0.6\) |
| \(2 \times 0.1 + 3 \times 0.2 + 4 \times 0.1 + 5a + 6b = 4.55\) oe | M1 | Allow equivalents e.g. \(5a + 6b = 3.35\) |
| \(5(0.6 - b) + 6b = 3.35 \Rightarrow a = 0.25\) or \(b = 0.35\) | M1 | Correct method to eliminate \(a\) or \(b\). May see \(a = c - 0.4\) or \(b = 1 - c\) |
| \(c = 0.4 + \text{"0.25"}\) or \(c = 1 - \text{"0.35"}\) | M1 | Complete method for finding \(c\) (condone any \(a\) and \(b\)) |
| \(c = 0.65\) oe | A1 | 0.65 oe |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(0.9 \times 0.75 \times 0.1\) | M1 | For product of 3 probabilities |
| \(= 0.0675\) | A1 | 0.0675 or exact fraction \(\dfrac{27}{400}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Identifying that if Jessie scores 2, Pabel has no spin | M1 | Identifying only one spin, or the 4 correct possibilities |
| \([0.10 \times 0 +] 0.2 \times 0.3 + 0.1 \times 0.15 + \text{"0.25"} \times 0.1 + \text{"0.35"} \times 0.2\) | M1 | At least 3 correct non-zero probability products ft their \(a\) and \(b\) (answer of 0.195 scores M0M1A0) |
| \(= 0.17\) | A1 | 0.17 |
# Question 5:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $E(R^2) = 2^2 \times 0.25 + 3^2 \times 0.3 + 4^2 \times 0.15 + 5^2 \times 0.1 + 6^2 \times 0.2 (= 15.8^*)$ | B1cso* | Correct calculation with all products seen (allow $1 + 2.7 + 2.4 + 2.5 + 7.2$). Figures may be seen in table before part (a). Condone missing addition signs if products seen in table |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\text{sd}(R) = \sqrt{15.8 - 3.7^2}$ | M1 | Use of formula including square root |
| $= \sqrt{2.11}$ | | |
| Standard deviation $= 1.4525\ldots$ awrt 1.45 | A1 | Correct answer with no working scores M1A1 |
## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| $d = 1$ | B1 | For 1 |
## Part (d)
| Answer | Mark | Guidance |
|--------|------|----------|
| $0.1 + 0.2 + 0.1 + a + b = 1$ oe | M1 | Allow equivalents e.g. $a + b = 0.6$ |
| $2 \times 0.1 + 3 \times 0.2 + 4 \times 0.1 + 5a + 6b = 4.55$ oe | M1 | Allow equivalents e.g. $5a + 6b = 3.35$ |
| $5(0.6 - b) + 6b = 3.35 \Rightarrow a = 0.25$ **or** $b = 0.35$ | M1 | Correct method to eliminate $a$ or $b$. May see $a = c - 0.4$ or $b = 1 - c$ |
| $c = 0.4 + \text{"0.25"}$ **or** $c = 1 - \text{"0.35"}$ | M1 | Complete method for finding $c$ (condone any $a$ and $b$) |
| $c = 0.65$ oe | A1 | 0.65 oe |
## Part (e)
| Answer | Mark | Guidance |
|--------|------|----------|
| $0.9 \times 0.75 \times 0.1$ | M1 | For product of 3 probabilities |
| $= 0.0675$ | A1 | 0.0675 or exact fraction $\dfrac{27}{400}$ |
## Part (f)
| Answer | Mark | Guidance |
|--------|------|----------|
| Identifying that if Jessie scores 2, Pabel has no spin | M1 | Identifying only one spin, or the 4 correct possibilities |
| $[0.10 \times 0 +] 0.2 \times 0.3 + 0.1 \times 0.15 + \text{"0.25"} \times 0.1 + \text{"0.35"} \times 0.2$ | M1 | At least 3 correct non-zero probability products ft their $a$ and $b$ (answer of 0.195 scores M0M1A0) |
| $= 0.17$ | A1 | 0.17 |\begin{enumerate}
\item A red spinner is designed so that the score $R$ is given by the following probability distribution.
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$r$ & 2 & 3 & 4 & 5 & 6 \\
\hline
$\mathrm { P } ( R = r )$ & 0.25 & 0.3 & 0.15 & 0.1 & 0.2 \\
\hline
\end{tabular}
\end{center}
(a) Show that $\mathrm { E } \left( R ^ { 2 } \right) = 15.8$
Given also that $\mathrm { E } ( R ) = 3.7$\\
(b) find the standard deviation of $R$, giving your answer to 2 decimal places.
A yellow spinner is designed so that the score $Y$ is given by the probability distribution in the table below. The cumulative distribution function $\mathrm { F } ( y )$ is also given.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$y$ & 2 & 3 & 4 & 5 & 6 \\
\hline
$\mathrm { P } ( Y = y )$ & 0.1 & 0.2 & 0.1 & $a$ & $b$ \\
\hline
$\mathrm {~F} ( y )$ & 0.1 & 0.3 & 0.4 & $c$ & $d$ \\
\hline
\end{tabular}
\end{center}
(c) Write down the value of $d$
Given that $\mathrm { E } ( Y ) = 4.55$\\
(d) find the value of $c$
Pabel and Jessie play a game with these two spinners.\\
Pabel uses the red spinner.\\
Jessie uses the yellow spinner.\\
They take turns to spin their spinner.\\
The winner is the first person whose spinner lands on the number 2 and the game ends. Jessie spins her spinner first.\\
(e) Find the probability that Jessie wins on her second spin.\\
(f) Calculate the probability that, in a game, the score on Pabel's first spin is the same as the score on Jessie's first spin.
\hfill \mbox{\textit{Edexcel S1 2022 Q5 [14]}}