| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2007 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Sampling without replacement |
| Difficulty | Standard +0.3 Part (i) is a straightforward combinatorial calculation using C(7,4) = 35. Part (ii) requires standard application of E(X) and Var(X) formulas with the given probability distribution—routine calculations with no conceptual challenges. This is slightly easier than average as it's mostly mechanical computation with provided data. |
| Spec | 2.04a Discrete probability distributions5.01a Permutations and combinations: evaluate probabilities5.02b Expectation and variance: discrete random variables |
| \(r\) | 0 | 1 | 2 | 3 | 4 |
| \(\mathrm { P } ( X = r )\) | 0 | \(\frac { 4 } { 35 }\) | \(\frac { 18 } { 35 }\) | \(\frac { 12 } { 35 }\) | \(\frac { 1 } { 35 }\) |
| Answer | Marks | Guidance |
|---|---|---|
| or \(P(\text{all 4 correct}) = \frac{1}{{}^7C_4} = \frac{1}{35}\) | M1 for fractions or \({}^7C_4\) seen, A1 NB answer given | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(X) = 1\times\frac{4}{35} + 2\times\frac{18}{35} + 3\times\frac{12}{35} + 4\times\frac{1}{35} = \frac{80}{35} = 2\frac{2}{7} = 2.29\) | M1 for \(\Sigma rp\) (at least 3 terms correct), A1 CAO | |
| \(E(X^2) = 1\times\frac{4}{35} + 4\times\frac{18}{35} + 9\times\frac{12}{35} + 16\times\frac{1}{35} = \frac{200}{35} = 5.714\) | M1 for \(\Sigma x^2p\) (at least 3 terms correct) | |
| \(\text{Var}(X) = \frac{200}{35} - \left(\frac{80}{35}\right)^2 = \frac{24}{49} = 0.490\) (to 3 s.f.) | M1dep for their \(E(X)^2\), A1 FT their \(E(X)\) provided \(\text{Var}(X) > 0\) | 5 marks |
# Question 6:
## Part (i)
Either $P(\text{all 4 correct}) = \frac{4}{7}\times\frac{3}{6}\times\frac{2}{5}\times\frac{1}{4} = \frac{1}{35}$
or $P(\text{all 4 correct}) = \frac{1}{{}^7C_4} = \frac{1}{35}$ | M1 for fractions or ${}^7C_4$ seen, A1 **NB answer given** | 2 marks
## Part (ii)
$E(X) = 1\times\frac{4}{35} + 2\times\frac{18}{35} + 3\times\frac{12}{35} + 4\times\frac{1}{35} = \frac{80}{35} = 2\frac{2}{7} = 2.29$ | M1 for $\Sigma rp$ (at least 3 terms correct), A1 CAO |
$E(X^2) = 1\times\frac{4}{35} + 4\times\frac{18}{35} + 9\times\frac{12}{35} + 16\times\frac{1}{35} = \frac{200}{35} = 5.714$ | M1 for $\Sigma x^2p$ (at least 3 terms correct) |
$\text{Var}(X) = \frac{200}{35} - \left(\frac{80}{35}\right)^2 = \frac{24}{49} = 0.490$ (to 3 s.f.) | M1dep for their $E(X)^2$, A1 FT their $E(X)$ provided $\text{Var}(X) > 0$ | 5 marks
---6 In a phone-in competition run by a local radio station, listeners are given the names of 7 local personalities and are told that 4 of them are in the studio. Competitors phone in and guess which 4 are in the studio.\\
(i) Show that the probability that a randomly selected competitor guesses all 4 correctly is $\frac { 1 } { 35 }$.
Let $X$ represent the number of correct guesses made by a randomly selected competitor. The probability distribution of $X$ is shown in the table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$r$ & 0 & 1 & 2 & 3 & 4 \\
\hline
$\mathrm { P } ( X = r )$ & 0 & $\frac { 4 } { 35 }$ & $\frac { 18 } { 35 }$ & $\frac { 12 } { 35 }$ & $\frac { 1 } { 35 }$ \\
\hline
\end{tabular}
\end{center}
(ii) Find the expectation and variance of $X$.
\hfill \mbox{\textit{OCR MEI S1 2007 Q6 [7]}}