| 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.\\
(i) 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:
$$\text { elephant ride } \frac { 6 } { 10 } , \quad \text { camel ride } \frac { 7 } { 10 } , \quad \text { jungle tractor ride } \frac { 8 } { 10 } .$$
(ii) 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.\\
(iii) Find the probability that he is frightened.\\
(iv) Given that Ravi is not frightened, find the probability that he went on the camel ride.
\hfill \mbox{\textit{CAIE S1 2009 Q5 [9]}}