| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2008 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tree Diagrams |
| Type | Probability distribution from tree |
| Difficulty | Moderate -0.8 This is a straightforward tree diagram question with independent events (constant 50% probability). Students must draw the tree, read off probabilities using the multiplication rule (powers of 1/2), and calculate an expectation. All steps are routine applications of standard S1 techniques with no conceptual challenges or novel problem-solving required. |
| Spec | 2.03b Probability diagrams: tree, Venn, sample space2.04a Discrete probability distributions |
| \(x\) | 0 | 1 | 2 | 3 | 4 |
| \(\mathrm { P } ( X = x )\) | \(\frac { 1 } { 4 }\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Tree diagram with 4 or 5 pairs A and U, probabilities 0.5 | M1 | 4 or 5 pairs A and U seen no extra bits but condone (0,1) branches after any or all As |
| Exactly 4 pairs of A and U, must be labelled | A1 | |
| Correct diagram with all probs correct | A1 3 | Allow A1ft for 4 correct pairs and (0,1) branch(es) or A1ft for 5 correct pairs and no (0,1) branch(es) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(X=0) = \frac{1}{2}\) | B1 | \(P(0)\) correct |
| \(P(X=2) = \frac{1}{4}\) | B1 | \(P(2)\) correct |
| \(P(X=3) = \frac{1}{8}\) | B1 | \(P(3)\) correct |
| \(P(X=4) = \frac{1}{16},\ P(X=4)=\frac{1}{16}\) | B1 4 | \(P(4)\) correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(E(X) = \frac{15}{16}\) (0.938 or 0.9375) | M1 | Attempt at \(\Sigma(xp)\) only with no other numbers |
| A1 2 | Correct answer |
## Question 6:
### Part (i)
| Answer | Mark | Guidance |
|--------|------|----------|
| Tree diagram with 4 or 5 pairs A and U, probabilities 0.5 | M1 | 4 or 5 pairs A and U seen no extra bits but condone (0,1) branches after any or all As |
| Exactly 4 pairs of A and U, must be labelled | A1 | |
| Correct diagram with all probs correct | A1 **3** | Allow A1ft for 4 correct pairs and (0,1) branch(es) or A1ft for 5 correct pairs and no (0,1) branch(es) |
### Part (ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(X=0) = \frac{1}{2}$ | B1 | $P(0)$ correct |
| $P(X=2) = \frac{1}{4}$ | B1 | $P(2)$ correct |
| $P(X=3) = \frac{1}{8}$ | B1 | $P(3)$ correct |
| $P(X=4) = \frac{1}{16},\ P(X=4)=\frac{1}{16}$ | B1 **4** | $P(4)$ correct |
### Part (iii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $E(X) = \frac{15}{16}$ (0.938 or 0.9375) | M1 | Attempt at $\Sigma(xp)$ only with no other numbers |
| | A1 **2** | Correct answer |
---6 Every day Eduardo tries to phone his friend. Every time he phones there is a $50 \%$ chance that his friend will answer. If his friend answers, Eduardo does not phone again on that day. If his friend does not answer, Eduardo tries again in a few minutes' time. If his friend has not answered after 4 attempts, Eduardo does not try again on that day.\\
(i) Draw a tree diagram to illustrate this situation.\\
(ii) Let $X$ be the number of unanswered phone calls made by Eduardo on a day. Copy and complete the table showing the probability distribution of $X$.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & 0 & 1 & 2 & 3 & 4 \\
\hline
$\mathrm { P } ( X = x )$ & & $\frac { 1 } { 4 }$ & & & \\
\hline
\end{tabular}
\end{center}
(iii) Calculate the expected number of unanswered phone calls on a day.
\hfill \mbox{\textit{CAIE S1 2008 Q6 [9]}}