| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2014 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Independent binomial samples with compound probability |
| Difficulty | Standard +0.3 Part (i) requires systematic enumeration of ways four dice sum to 5 (only 1+1+1+2 and permutations), which is straightforward but requires care. Part (ii) is a standard binomial probability application using the result from (i). Both parts involve routine techniques with no novel insight, making this slightly easier than average for A-level. |
| Spec | 2.04a Discrete probability distributions2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities5.01a Permutations and combinations: evaluate probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| \[\text{Prob} = \frac{1}{6} \times \frac{1}{6} \times \frac{1}{6} \times \frac{1}{6} = \frac{1}{324} (0.00309)\] | M1, M1, A1 3 | One of \(1 \mid 1 \mid 2\) seen or Mult a prob by 4 or \((\frac{1}{6})^4 \times\) integer \(k \geq 1\) seen or Correct answer |
| (ii) \[P(1,2) = ^7C_1 \times (1/324)(323/324)^6 + ^7C_4(1/324)^4(323/324)^3 = 0.0214\] | M1, M1, M1, A1 4 | Bin term \(^7C_p(q)^{-x} \cdot {}^xC_r\), \(0.99 \leq p + q \leq 1\) or Using their \(p\) from (i) in a bin term or Correct unsimplified answer or Correct answer |
**(i)** One of $1 \mid 1 \mid 2$, or $1 \mid 2 \mid 1$, or $2 \mid 1 \mid 1$, or $2 \mid 1 \mid 1$
$$\text{Prob} = \frac{1}{6} \times \frac{1}{6} \times \frac{1}{6} \times \frac{1}{6} = \frac{1}{324} (0.00309)$$ | M1, M1, A1 3 | One of $1 \mid 1 \mid 2$ seen or Mult a prob by 4 or $(\frac{1}{6})^4 \times$ integer $k \geq 1$ seen or Correct answer
**(ii)** $$P(1,2) = ^7C_1 \times (1/324)(323/324)^6 + ^7C_4(1/324)^4(323/324)^3 = 0.0214$$ | M1, M1, M1, A1 4 | Bin term $^7C_p(q)^{-x} \cdot {}^xC_r$, $0.99 \leq p + q \leq 1$ or Using their $p$ from (i) in a bin term or Correct unsimplified answer or Correct answer
---\begin{enumerate}[label=(\roman*)]
\item Four fair six-sided dice, each with faces marked 1, 2, 3, 4, 5, 6, are thrown. Find the probability that the numbers shown on the four dice add up to 5. [3]
\item Four fair six-sided dice, each with faces marked 1, 2, 3, 4, 5, 6, are thrown on 7 occasions. Find the probability that the numbers shown on the four dice add up to 5 on exactly 1 or 2 of the 7 occasions. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2014 Q3 [7]}}