| Exam Board | Edexcel |
|---|---|
| Module | Paper 3 (Paper 3) |
| Year | 2020 |
| Session | October |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | One unknown from sum constraint only |
| Difficulty | Standard +0.3 This is a straightforward probability question requiring: (a) summing fractions to find k using ΣP=1, (b) listing cases where D₁+D₂=80 and multiplying independent probabilities, (c) determining which values of d create valid quadrilaterals with smallest angle >50°. All steps are routine applications of basic probability rules with no novel insight required, making it slightly easier than average. |
| Spec | 2.04a Discrete probability distributions2.04c Calculate binomial probabilities |
| \(d\) | 10 | 20 | 30 | 40 | 50 |
| \(\mathrm { P } ( D = d )\) | \(\frac { k } { 10 }\) | \(\frac { k } { 20 }\) | \(\frac { k } { 30 }\) | \(\frac { k } { 40 }\) | \(\frac { k } { 50 }\) |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(\frac{k}{10} + \frac{k}{20} + \frac{k}{30} + \frac{k}{40} + \frac{k}{50} = 1\) or \(\frac{1}{600}(60k + 30k + 20k + 15k + 12k) = 1\). So \(k = \frac{600}{137}\) \((*)\) | M1; A1cso |
| (b) | (Cases are:) \(D_1 = 30, D_x = 50\) and \(D_x = 50, D_x = 30\) and \(D_x = 40, D_x = 40\). \(\text{P}(D_1 + D_2 = 80) = \frac{k}{50} \times \frac{k}{30} \times 2 + \left(\frac{k}{40}\right)^2\) = \(0.0375619...\) awrt \(\mathbf{0.0376}\) | M1; M1; A1 |
| (c) | Angles are: \(a, a+d, a+2d, a+3d\). \(S_4 = a + (a+d) + (a+2d) + (a+3d) = 360\). \(2a + 3d = 180\) (o.e.). Smallest angle is \(a > 50\) consider cases: \(d = 10\) so \(a = 75\) or \(d = 20\) so \(a = 60\) [\(d = 30\) gives \(a = 45\) no good]. \(\text{P}(D = 10 \text{ or } 20) = \frac{3k}{20} = \frac{90}{137}\) | M1; M1; A1; M1; A1 |
| (a) | $\frac{k}{10} + \frac{k}{20} + \frac{k}{30} + \frac{k}{40} + \frac{k}{50} = 1$ or $\frac{1}{600}(60k + 30k + 20k + 15k + 12k) = 1$. So $k = \frac{600}{137}$ $(*)$ | M1; A1cso | M1 for clear use of sum of probabilities = 1 (all terms seen). A1cso (*) M1 scored and no incorrect working seen. Assume $k = \frac{600}{137}$ to score the final A1 they must have a final comment "$\therefore k = \frac{600}{137}$". |
|---|---|---|---|
| (b) | (Cases are:) $D_1 = 30, D_x = 50$ and $D_x = 50, D_x = 30$ and $D_x = 40, D_x = 40$. $\text{P}(D_1 + D_2 = 80) = \frac{k}{50} \times \frac{k}{30} \times 2 + \left(\frac{k}{40}\right)^2$ = $0.0375619...$ awrt $\mathbf{0.0376}$ | M1; M1; A1 | 1st M1 for selecting at least 2 of the relevant cases (may be implied by their correct probs) e.g. allow 30, 50 and 50, 30. i.e. $D_1$ and $D_2$ labels not required. 2nd M1 for using the model to obtain a correct expression for two different probabilities. May use letter $k$ or their value for $k$. Allow for $\frac{k}{50} \times \frac{k}{30} + \left(\frac{k}{40}\right)^2$ or $2 \times \left(\frac{k}{50} \times \frac{k}{30} + \left(\frac{k}{40}\right)^2\right)$. A1 for awrt 0.0376 (exact fraction is $\frac{705}{18769}$). |
| (c) | Angles are: $a, a+d, a+2d, a+3d$. $S_4 = a + (a+d) + (a+2d) + (a+3d) = 360$. $2a + 3d = 180$ (o.e.). Smallest angle is $a > 50$ consider cases: $d = 10$ so $a = 75$ or $d = 20$ so $a = 60$ [$d = 30$ gives $a = 45$ no good]. $\text{P}(D = 10 \text{ or } 20) = \frac{3k}{20} = \frac{90}{137}$ | M1; M1; A1; M1; A1 | 1st M1 for recognising the 4 angles and finding expressions in terms of $d$ and their $a$. 2nd M1 for using property of quad with these 4 angles (equation can be un-simplified). Allow these two marks for use of a (possible) value of $d$ (3 cases for A1) e.g. $a + a + 10 + a + 20 + a + 30 = 360$ (If at least 3 cases seen allow A1 for e.g. $4a = 300$). 1st A1 for $2a + 3d = 180$ condition (o.e.) [Must be in the form $pa + qd = N$]. 3rd M1 for examining cases and getting $d = 10$ and $d = 20$ only. 2nd A1 for $\frac{90}{137}$ or exact equivalent. The correct answer and no obviously incorrect working will score 5/5. A final answer of awrt 0.657 (0.65693...) with no obviously incorrect working scores 4/5. |\begin{enumerate}
\item The discrete random variable $D$ has the following probability distribution
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$d$ & 10 & 20 & 30 & 40 & 50 \\
\hline
$\mathrm { P } ( D = d )$ & $\frac { k } { 10 }$ & $\frac { k } { 20 }$ & $\frac { k } { 30 }$ & $\frac { k } { 40 }$ & $\frac { k } { 50 }$ \\
\hline
\end{tabular}
\end{center}
where $k$ is a constant.\\
(a) Show that the value of $k$ is $\frac { 600 } { 137 }$
The random variables $D _ { 1 }$ and $D _ { 2 }$ are independent and each have the same distribution as $D$.\\
(b) Find $\mathrm { P } \left( D _ { 1 } + D _ { 2 } = 80 \right)$
Give your answer to 3 significant figures.
A single observation of $D$ is made.\\
The value obtained, $d$, is the common difference of an arithmetic sequence.\\
The first 4 terms of this arithmetic sequence are the angles, measured in degrees, of quadrilateral $Q$\\
(c) Find the exact probability that the smallest angle of $Q$ is more than $50 ^ { \circ }$
\hfill \mbox{\textit{Edexcel Paper 3 2020 Q4 [10]}}