| Exam Board | OCR MEI |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2021 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | One unknown from sum constraint only |
| Difficulty | Moderate -0.3 This is a straightforward probability distribution question requiring students to use the sum-to-one constraint to form and solve a quadratic equation, then identify the mode. While it involves algebraic manipulation of a quadratic, the conceptual demand is low—it's a standard textbook exercise with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04a Discrete probability distributions |
| \(x\) | 0 | 1 | 2 | 3 |
| \(\mathrm { P } ( X = x )\) | \(3 p ^ { 2 }\) | \(0.5 p ^ { 2 } + 2 p\) | \(1.5 p\) | \(1.5 p ^ { 2 } + 0.5 p\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(3p^2 + 0.5p^2 + 2p + 1.5p + 1.5p^2 + 0.5p = 1\) | M1 | Allow if only 3 probs added, or if \(= 1\) omitted |
| \(5p^2 + 4p - 1 = 0\) | A1 | Allow if one coeff or one sign wrong |
| \(p = 0.2\) or \(-1\) BC | A1 | |
| \(p = 0.2\) only | A1 | |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use of their \(p\) to calculate at least two probabilities | B1 | Their \(p\) must be \(< 1\) |
| Mode is 1 | B1 | WWW. NB \(0.12, 0.42, 0.3, 0.16\) |
| [2] |
## Question 6:
### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $3p^2 + 0.5p^2 + 2p + 1.5p + 1.5p^2 + 0.5p = 1$ | M1 | Allow if only 3 probs added, or if $= 1$ omitted |
| $5p^2 + 4p - 1 = 0$ | A1 | Allow if one coeff or one sign wrong |
| $p = 0.2$ or $-1$ **BC** | A1 | |
| $p = 0.2$ only | A1 | |
| **[4]** | | |
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use of their $p$ to calculate at least two probabilities | B1 | Their $p$ must be $< 1$ |
| Mode is 1 | B1 | WWW. NB $0.12, 0.42, 0.3, 0.16$ |
| **[2]** | | |
---6 The probability distribution for the discrete random variable $X$ is shown below.
\begin{center}
\begin{tabular}{ | l | l | l | l | l | }
\hline
$x$ & 0 & 1 & 2 & 3 \\
\hline
$\mathrm { P } ( X = x )$ & $3 p ^ { 2 }$ & $0.5 p ^ { 2 } + 2 p$ & $1.5 p$ & $1.5 p ^ { 2 } + 0.5 p$ \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Determine the value of $p$.
\item Determine the modal value of $X$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI AS Paper 2 2021 Q6 [6]}}