| Exam Board | OCR MEI |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | One unknown from sum constraint only |
| Difficulty | Easy -1.3 This is a straightforward probability distribution question requiring only basic recall and routine calculations: (a) uses the sum-to-1 constraint to find q algebraically, (b) is simple addition, (c) applies independence with P(X=1)×P(X=2)×2, and (d) is a standard binomial probability calculation. All parts are textbook exercises with no problem-solving or novel insight required. |
| Spec | 2.04a Discrete probability distributions2.04c Calculate binomial probabilities |
| \(x\) | 1 | 2 | 3 | 4 | 5 |
| \(\mathrm { P } ( \mathrm { X } = \mathrm { x } )\) | 0.1 | 0.3 | \(q\) | \(2 q\) | \(3 q\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(0.1 + 0.3 + q + 2q + 3q = 1\) | M1 | Setting sum of values equal to 1 |
| \(q = 0.1\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \((0.1 + 0.3 + 0.1 + 0.2 =)\ 0.7\) | B1 | Or \(1 - p(X=5)\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(0.1 \times 0.3\) seen or \(0.03\) seen | M1 | |
| \(0.1 \times 0.3 + 0.1 \times 0.3 = 0.06\) | A1 | \(0.06\) o.e. \(\frac{6}{100}\) or \(\frac{3}{50}\) etc |
| Answer | Marks | Guidance |
|---|---|---|
| \(0.098314\ldots\) correct to 2 or more sf | B1 | By using e.g. \(X \sim B(50, 0.3)\) and finding \(P(X=17)\) |
## Question 9:
### Part (a):
$0.1 + 0.3 + q + 2q + 3q = 1$ | **M1** | Setting sum of values equal to 1
$q = 0.1$ | **A1** |
### Part (b):
$(0.1 + 0.3 + 0.1 + 0.2 =)\ 0.7$ | **B1** | Or $1 - p(X=5)$
### Part (c):
$0.1 \times 0.3$ seen or $0.03$ seen | **M1** |
$0.1 \times 0.3 + 0.1 \times 0.3 = 0.06$ | **A1** | $0.06$ o.e. $\frac{6}{100}$ or $\frac{3}{50}$ etc
### Part (d):
$0.098314\ldots$ correct to 2 or more sf | **B1** | By using e.g. $X \sim B(50, 0.3)$ and finding $P(X=17)$
---9 The table shows the probability distribution for the discrete random variable $X$.
\begin{center}
\begin{tabular}{ | l | l | l | l | l | l | }
\hline
$x$ & 1 & 2 & 3 & 4 & 5 \\
\hline
$\mathrm { P } ( \mathrm { X } = \mathrm { x } )$ & 0.1 & 0.3 & $q$ & $2 q$ & $3 q$ \\
\hline
\end{tabular}
\end{center}
You are given that $q$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Determine the value of $q$.
\item Calculate $\mathrm { P } ( X \leqslant 4 )$.
Two independent values of $X$ are taken.
\item Determine the probability that the sum of the two values is 3 .
Fifty independent values of $X$ are taken.
\item Find the probability that a value of 2 occurs exactly 17 times.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI AS Paper 2 2023 Q9 [6]}}