| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Verify probability from independent trials |
| Difficulty | Standard +0.3 This is a straightforward probability question involving independent trials with one biased coin. Parts (i)-(ii) require basic probability calculations with clear guidance ('show that'), part (v) uses standard expectation formulas, and part (vi) applies the given distribution. While multi-part, each step follows routine procedures without requiring novel insight or complex problem-solving. |
| Spec | 2.04a Discrete probability distributions2.04b Binomial distribution: as model B(n,p) |
| \(r\) | 0 | 1 | ||||
| \(\mathrm { P } ( X = r )\) | 0.025 | 0.1375 | 0.3 | 0.325 | 0.175 | 0.0375 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| 'Negative' or 'very slight negative' | E1 [1] | E0 for symmetrical; E1 for (very slight) negative skewness even if also mention symmetrical; ignore any reference to unimodal |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(X) = (0\times0.025) + (1\times0.1375) + (2\times0.3) + (3\times0.325) + (4\times0.175) + (5\times0.0375) = 2.6\) | M1, A1 | M1 for \(\Sigma rp\) (at least 3 terms correct); CAO |
| \(E(X^2) = (0\times0.025) + (1\times0.1375) + (4\times0.3) + (9\times0.325) + (16\times0.175) + (25\times0.075) = 8\) | M1* | For \(\Sigma r^2p\) (at least 3 terms correct) |
| \(\text{Var}(X) = 8 - 2.6^2 = 1.24\) | M1* dep, A1 [5] | M1* for their \(E(X)^2\); A1 FT their \(E(X)\) provided \(\text{Var}(X) > 0\); USE of \(E(X-\mu)^2\) gets M1 for attempt at \((x-\mu)^2\), should see \((-2.6)^2, (-1.6)^2, (-0.6)^2, 0.4^2, 1.4^2, 2.4^2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(\text{Total of } 3) = (3\times0.325\times0.025^2) + (6\times0.3\times0.1375\times0.025)\) | M1 | For decimal part of first term \(0.325\times0.025^2\) |
| \(0.1375^3 = 3\times0.000203 + 6\times0.001031 + 0.002600\) | M1 | For decimal part of second term \(0.3\times0.1375\times0.025\) |
| \(= 0.000609 + 0.006188 + 0.002600 = 0.00940\) | M1, A1 [4] | M1 for third term – ignore extra coefficient; CAO: AWRT 0.0094; allow 0.009 with working |
# Question 1:
## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| 'Negative' or 'very slight negative' | E1 [1] | E0 for symmetrical; E1 for (very slight) negative skewness even if also mention symmetrical; ignore any reference to unimodal |
## Part (v)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(X) = (0\times0.025) + (1\times0.1375) + (2\times0.3) + (3\times0.325) + (4\times0.175) + (5\times0.0375) = 2.6$ | M1, A1 | M1 for $\Sigma rp$ (at least 3 terms correct); CAO |
| $E(X^2) = (0\times0.025) + (1\times0.1375) + (4\times0.3) + (9\times0.325) + (16\times0.175) + (25\times0.075) = 8$ | M1* | For $\Sigma r^2p$ (at least 3 terms correct) |
| $\text{Var}(X) = 8 - 2.6^2 = 1.24$ | M1* dep, A1 [5] | M1* for their $E(X)^2$; A1 FT their $E(X)$ provided $\text{Var}(X) > 0$; USE of $E(X-\mu)^2$ gets M1 for attempt at $(x-\mu)^2$, should see $(-2.6)^2, (-1.6)^2, (-0.6)^2, 0.4^2, 1.4^2, 2.4^2$ |
## Part (vi)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(\text{Total of } 3) = (3\times0.325\times0.025^2) + (6\times0.3\times0.1375\times0.025)$ | M1 | For decimal part of first term $0.325\times0.025^2$ |
| $0.1375^3 = 3\times0.000203 + 6\times0.001031 + 0.002600$ | M1 | For decimal part of second term $0.3\times0.1375\times0.025$ |
| $= 0.000609 + 0.006188 + 0.002600 = 0.00940$ | M1, A1 [4] | M1 for third term – ignore extra coefficient; CAO: AWRT 0.0094; allow 0.009 with working |
---1 Yasmin has 5 coins. One of these coins is biased with P (heads) $= 0.6$. The other 4 coins are fair. She tosses all 5 coins once and records the number of heads, $X$.\\
(i) Show that $\mathrm { P } ( X = 0 ) = 0.025$.\\
(ii) Show that $\mathrm { P } ( X = 1 ) = 0.1375$.
The table shows the probability distribution of $X$.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
$r$ & 0 & 1 & & & & \\
\hline
$\mathrm { P } ( X = r )$ & 0.025 & 0.1375 & 0.3 & 0.325 & 0.175 & 0.0375 \\
\hline
\end{tabular}
\end{center}
(iii) Draw a vertical line chart to illustrate the probability distribution.\\
(iv) Comment on the skewness of the distribution.\\
(v) Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.\\
(vi) Yasmin tosses the 5 coins three times. Find the probability that the total number of heads is 3 .
\hfill \mbox{\textit{OCR MEI S1 Q1 [18]}}