| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Verify probability from combinatorial selection |
| Difficulty | Moderate -0.3 Part (i) is a straightforward application of combinations (choosing 4 from 7), requiring only basic counting principles. Part (ii) involves routine calculation of E(X) and Var(X) from a given probability distribution using standard formulas. While multi-step, both parts are standard textbook exercises with no novel insight required, making this slightly easier than average. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.02a Discrete probability distributions: general5.02b 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 |
|---|---|---|
| \(P(\text{all 4 correct}) = \frac{4}{7}\times\frac{3}{6}\times\frac{2}{5}\times\frac{1}{4} = \frac{1}{35}\) | M1 | for fractions, or \(^7C_4\) seen |
| or \(P(\text{all 4 correct}) = \frac{1}{^7C_4} = \frac{1}{35}\) | A1 | NB answer given |
| 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 \(\text{Var}(X) > 0\) |
## Question 3:
### Part (i)
| $P(\text{all 4 correct}) = \frac{4}{7}\times\frac{3}{6}\times\frac{2}{5}\times\frac{1}{4} = \frac{1}{35}$ | M1 | for fractions, or $^7C_4$ seen |
| or $P(\text{all 4 correct}) = \frac{1}{^7C_4} = \frac{1}{35}$ | A1 | **NB answer given** |
### 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 $\text{Var}(X) > 0$ |
---3 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 Q3 [7]}}