| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2009 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tree Diagrams |
| Type | Conditional probability from tree |
| Difficulty | Moderate -0.8 This is a straightforward tree diagram question requiring basic probability calculations (coin tosses, total probability theorem, and Bayes' theorem). All steps are routine applications of standard formulas with no conceptual challenges beyond AS-level content. |
| Spec | 2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space2.03d Calculate conditional probability: from first principles |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(E) = \frac{1}{4},\ P(C) = \frac{1}{4},\ P(JT) = \frac{1}{2}\) | B1 | \(\frac{1}{4}\), \(\frac{1}{4}\), and \(\frac{1}{2}\) seen oe |
| B1 [2] | 3 evaluated probabilities correctly associated |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Tree diagram with E, C, JT then F on appropriate shape | M1 | E, C, JT then F on appropriate shape |
| All probabilities and labels showing and correct, ft their (i) | A1ft [2] | All probs and labels showing and correct, ft their (i) if \(\Sigma p = 1\). If nothing seen in part (i) then give M1 A1ft bod provided their \(\Sigma p = 1\). No retrospective marking. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(F) = \left(\frac{1}{4} \times \frac{6}{10}\right) + \left(\frac{1}{4} \times \frac{7}{10}\right) + \left(\frac{1}{2} \times \frac{8}{10}\right)\) | M1 | Summing 3 appropriate two-factor products provided \(\Sigma p = 1\) |
| \(= \frac{29}{40}\) (0.725) | B1 [2] | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(C \mid NF) = \frac{P(C \cap NF)}{P(NF)}\) | B1ft | \(1 - \frac{29}{40}\) seen in denom, ft \(1\) – their (iii) |
| \(= \frac{3/40}{(1 - 29/40)}\) | M1 | Attempt at conditional prob with their \(C \cap F\) or \(C \cap NF\) in numerator |
| \(= \frac{3}{11}\) (0.273) | A1 [3] | Correct answer |
| OR using ratios \(3/(4+3+4)\) |
## Question 5:
### Part (i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(E) = \frac{1}{4},\ P(C) = \frac{1}{4},\ P(JT) = \frac{1}{2}$ | B1 | $\frac{1}{4}$, $\frac{1}{4}$, and $\frac{1}{2}$ seen oe |
| | B1 **[2]** | 3 evaluated probabilities correctly associated |
### Part (ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Tree diagram with E, C, JT then F on appropriate shape | M1 | E, C, JT then F on appropriate shape |
| All probabilities and labels showing and correct, ft their **(i)** | A1ft **[2]** | All probs and labels showing and correct, ft their **(i)** if $\Sigma p = 1$. If nothing seen in part **(i)** then give M1 A1ft bod provided their $\Sigma p = 1$. No retrospective marking. |
### Part (iii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(F) = \left(\frac{1}{4} \times \frac{6}{10}\right) + \left(\frac{1}{4} \times \frac{7}{10}\right) + \left(\frac{1}{2} \times \frac{8}{10}\right)$ | M1 | Summing 3 appropriate two-factor products provided $\Sigma p = 1$ |
| $= \frac{29}{40}$ (0.725) | B1 **[2]** | Correct answer |
### Part (iv)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(C \mid NF) = \frac{P(C \cap NF)}{P(NF)}$ | B1ft | $1 - \frac{29}{40}$ seen in denom, ft $1$ – their **(iii)** |
| $= \frac{3/40}{(1 - 29/40)}$ | M1 | Attempt at conditional prob with their $C \cap F$ or $C \cap NF$ in numerator |
| $= \frac{3}{11}$ (0.273) | A1 **[3]** | Correct answer |
| OR using ratios $3/(4+3+4)$ | | |
---5 At a zoo, rides are offered on elephants, camels and jungle tractors. Ravi has money for only one ride. To decide which ride to choose, he tosses a fair coin twice. If he gets 2 heads he will go on the elephant ride, if he gets 2 tails he will go on the camel ride and if he gets 1 of each he will go on the jungle tractor ride.\\
\begin{enumerate}[label=(\roman*)]
\item Find the probabilities that he goes on each of the three rides.
The probabilities that Ravi is frightened on each of the rides are as follows:
$$\text { elephant ride } \frac { 6 } { 10 } , \quad \text { camel ride } \frac { 7 } { 10 } , \quad \text { jungle tractor ride } \frac { 8 } { 10 } .$$
\item Draw a fully labelled tree diagram showing the rides that Ravi could take and whether or not he is frightened.
Ravi goes on a ride.
\item Find the probability that he is frightened.
\item Given that Ravi is not frightened, find the probability that he went on the camel ride.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2009 Q5 [9]}}