| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2017 |
| Session | March |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Construct probability distribution from scenario |
| Difficulty | Moderate -0.3 This is a straightforward probability distribution question requiring systematic enumeration of outcomes from two discrete sets. Part (i) is guided, part (ii) involves routine calculation of all possible sums and their probabilities, and part (iii) applies basic conditional probability. While multi-part, each step uses standard S1 techniques without requiring novel insight. |
| Spec | 2.03d Calculate conditional probability: from first principles2.04a Discrete probability distributions |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(2) = P(0,2) = \frac{2}{10} \times \frac{4}{6}\) | M1 | Mult 2 probs seen (or complete listing of all options) |
| \(= \frac{2}{15}\) | A1 AG | Correct answer legit obtained |
| Answer | Marks | Guidance |
|---|---|---|
| \(x\) | \(0\) | \(1\) |
| \(P(X=x)\) | \(\frac{2}{30}\) | \(\frac{5}{30}\) |
| B1 | Correct values for \(x\) in table. Any additional values must have \(P(x)=0\) stated | |
| B1 | One correct prob other than \(P(2)\) or \(P(3)\) | |
| B1 | Correct \(P(3)\) | |
| B1 | All correct |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(A1 | \text{Sum }3) = \dfrac{P(A1 \cap \text{Sum3})}{P(\text{Sum3})} = \dfrac{5/10 \times 4/6}{13/30}\) | M1 |
| M1 | Their \(P(3)\) from (ii) as num or denom of a fraction | |
| \(= \frac{10}{13}\ (0.769)\) | A1 |
# Question 6(i):
$P(2) = P(0,2) = \frac{2}{10} \times \frac{4}{6}$ | M1 | Mult 2 probs seen (or complete listing of all options)
$= \frac{2}{15}$ | A1 AG | Correct answer legit obtained
**Total: 2**
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# Question 6(ii):
| $x$ | $0$ | $1$ | $2$ | $3$ | $5$ |
|---|---|---|---|---|---|
| $P(X=x)$ | $\frac{2}{30}$ | $\frac{5}{30}$ | $\frac{4}{30}$ | $\frac{13}{30}$ | $\frac{6}{30}$ |
| B1 | Correct values for $x$ in table. Any additional values must have $P(x)=0$ stated
| B1 | One correct prob other than $P(2)$ or $P(3)$
| B1 | Correct $P(3)$
| B1 | All correct
**Total: 4**
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# Question 6(iii):
$P(A1|\text{Sum }3) = \dfrac{P(A1 \cap \text{Sum3})}{P(\text{Sum3})} = \dfrac{5/10 \times 4/6}{13/30}$ | M1 | Attempt at $P(A1 \cap \text{Sum }3)$ as num or denom of a fraction, can be by counting
| M1 | Their $P(3)$ from (ii) as num or denom of a fraction
$= \frac{10}{13}\ (0.769)$ | A1 |
**Total: 3**
---6 Pack $A$ consists of ten cards numbered $0,0,1,1,1,1,1,3,3,3$. Pack $B$ consists of six cards numbered $0,0,2,2,2,2$. One card is chosen at random from each pack. The random variable $X$ is defined as the sum of the two numbers on the cards.\\
(i) Show that $\mathrm { P } ( X = 2 ) = \frac { 2 } { 15 }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{556a1cc2-47ef-4ef7-a8f6-42850c303531-08_59_1569_497_328}\\
(ii) Draw up the probability distribution table for $X$.\\
(iii) Given that $X = 3$, find the probability that the card chosen from pack $A$ is a 1 .\\
\hfill \mbox{\textit{CAIE S1 2017 Q6 [9]}}