| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2013 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Definitions |
| Type | Sequential events and tree diagrams |
| Difficulty | Moderate -0.8 This is a straightforward probability question involving conditional definitions and systematic enumeration. Parts (i) and (ii) guide students through the method, part (iii) requires routine calculation of remaining probabilities, and part (iv) tests basic understanding of exclusive events. The question requires careful organization but no novel insight—it's easier than average A-level work due to its structured guidance and standard techniques. |
| Spec | 2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space2.03c Conditional probability: using diagrams/tables2.04a Discrete probability distributions |
| \(x\) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| \(\mathrm { P } ( X = x )\) | \(\frac { 1 } { 8 }\) | \(\frac { 3 } { 16 }\) | \(\frac { 1 } { 8 }\) | \(\frac { 1 } { 16 }\) | \(\frac { 1 } { 32 }\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| If throw H then smallest score is 2; \(P(T,1) = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8}\) AG | B1, B1 [2] | Or equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P(3)\) from two dice \(= \frac{2}{16}\) seen | B1 | From \((1,2)\) and \((2,1)\) |
| \(P(H,3) = \frac{1}{2} \times \frac{2}{16} = \frac{2}{32}\); \(P(T,3) = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8}\) | M1, A1 | Summing \(P(H,3)\) and \(P(T,3)\); one correct |
| So \(P(3) = \frac{6}{32} = \frac{3}{16}\) AG | A1 [4] | Correct answer, must see clear reasoning |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(X\): 1, 2, 3, 4, 5, 6, 7, 8 | B1 | One correct prob |
| Prob: —, \(\frac{5}{32}\), —, \(\frac{7}{32}\), —, \(\frac{3}{32}\), —, — | B1 | A second correct prob |
| B1 [3] | A third correct prob |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P(Q \cap R) = 0\) or 'if you throw a tail you can't get a 7' | M1 | Stating \(P(Q \cap R) = 0\) or implying by words |
| Yes they are exclusive | A1dep [2] | Dep on previous M |
## Question 7:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| If throw H then smallest score is 2; $P(T,1) = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8}$ **AG** | B1, B1 **[2]** | Or equivalent |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(3)$ from two dice $= \frac{2}{16}$ seen | B1 | From $(1,2)$ and $(2,1)$ |
| $P(H,3) = \frac{1}{2} \times \frac{2}{16} = \frac{2}{32}$; $P(T,3) = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8}$ | M1, A1 | Summing $P(H,3)$ and $P(T,3)$; one correct |
| So $P(3) = \frac{6}{32} = \frac{3}{16}$ **AG** | A1 **[4]** | Correct answer, must see clear reasoning |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $X$: 1, 2, 3, 4, 5, 6, 7, 8 | B1 | One correct prob |
| Prob: —, $\frac{5}{32}$, —, $\frac{7}{32}$, —, $\frac{3}{32}$, —, — | B1 | A second correct prob |
| | B1 **[3]** | A third correct prob |
### Part (iv):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(Q \cap R) = 0$ or 'if you throw a tail you can't get a 7' | M1 | Stating $P(Q \cap R) = 0$ or implying by words |
| Yes they are exclusive | A1dep **[2]** | Dep on previous M |7 James has a fair coin and a fair tetrahedral die with four faces numbered 1, 2, 3, 4. He tosses the coin once and the die twice. The random variable $X$ is defined as follows.
\begin{itemize}
\item If the coin shows a head then $X$ is the sum of the scores on the two throws of the die.
\item If the coin shows a tail then $X$ is the score on the first throw of the die only.\\
(i) Explain why $X = 1$ can only be obtained by throwing a tail, and show that $\mathrm { P } ( X = 1 ) = \frac { 1 } { 8 }$.\\
(ii) Show that $\mathrm { P } ( X = 3 ) = \frac { 3 } { 16 }$.\\
(iii) Copy and complete the probability distribution table for $X$.
\end{itemize}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | }
\hline
$x$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
$\mathrm { P } ( X = x )$ & $\frac { 1 } { 8 }$ & & $\frac { 3 } { 16 }$ & & $\frac { 1 } { 8 }$ & & $\frac { 1 } { 16 }$ & $\frac { 1 } { 32 }$ \\
\hline
\end{tabular}
\end{center}
Event $Q$ is 'James throws a tail'. Event $R$ is 'the value of $X$ is 7'.\\
(iv) Determine whether events $Q$ and $R$ are exclusive. Justify your answer.
\hfill \mbox{\textit{CAIE S1 2013 Q7 [11]}}