| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2014 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conditional Probability |
| Type | Multi-stage game or match outcomes |
| Difficulty | Standard +0.3 This is a straightforward conditional probability problem with clearly defined states and transitions. Part (i) requires identifying two simple paths (RR or AA) and calculating their probabilities using given values. Part (ii) applies basic conditional probability formula. The structure is standard for S1 level with no conceptual surprises, making it slightly easier than average A-level questions. |
| Spec | 2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space2.03c Conditional probability: using diagrams/tables2.03d Calculate conditional probability: from first principles |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(P(RR) = 0.6 \times 0.7 = 0.42\) | B1 | Only 2 factors |
| \(P(AA) = 0.4 \times 0.75 = 0.3\) | B1 | Only 2 factors |
| \(P(\text{2 sets in match}) = 0.72\) | B1\(\checkmark\) [3] | ft previous answers |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\dfrac{P(A \text{ wins and 2 sets})}{P(\text{2 sets})} = \dfrac{P(AA)}{P(\text{2 sets})}\) | B1\(\checkmark\) | Correct numerator or correct denominator of a fraction ft their (i) |
| \(= \dfrac{0.3}{0.72} = \dfrac{5}{12}\ (0.417)\) | B1\(\checkmark\) [2] | Correct answer ft their or recovered AA / their recovered (i) |
## Question 3:
### Part (i):
| Working | Mark | Guidance |
|---------|------|----------|
| $P(RR) = 0.6 \times 0.7 = 0.42$ | B1 | Only 2 factors |
| $P(AA) = 0.4 \times 0.75 = 0.3$ | B1 | Only 2 factors |
| $P(\text{2 sets in match}) = 0.72$ | B1$\checkmark$ **[3]** | ft previous answers |
### Part (ii):
| Working | Mark | Guidance |
|---------|------|----------|
| $\dfrac{P(A \text{ wins and 2 sets})}{P(\text{2 sets})} = \dfrac{P(AA)}{P(\text{2 sets})}$ | B1$\checkmark$ | Correct numerator or correct denominator of a fraction ft their (i) |
| $= \dfrac{0.3}{0.72} = \dfrac{5}{12}\ (0.417)$ | B1$\checkmark$ **[2]** | Correct answer ft their or recovered AA / their recovered (i) |
---3 Roger and Andy play a tennis match in which the first person to win two sets wins the match. The probability that Roger wins the first set is 0.6 . For sets after the first, the probability that Roger wins the set is 0.7 if he won the previous set, and is 0.25 if he lost the previous set. No set is drawn.\\
(i) Find the probability that there is a winner of the match after exactly two sets.\\
(ii) Find the probability that Andy wins the match given that there is a winner of the match after exactly two sets.
\hfill \mbox{\textit{CAIE S1 2014 Q3 [5]}}