| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Combinations & Selection |
| Type | Probability distributions from selection |
| Difficulty | Moderate -0.8 This is a straightforward application of combinations to find probabilities. Part (i) requires calculating P(X=1) = C(3,1)×C(9,2)/C(12,3), which is routine computation. Part (ii) extends this to find P(X=0), P(X=2), P(X=3) using the same method. No conceptual difficulty beyond basic combination formulas and the definition of a probability distribution—easier than average A-level work. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P(1 \text{ T-shirt}) = \dfrac{^3C_1 \times ^9C_2}{^{12}C_3}\) | B1 | Correct numerator unsimplified |
| B1 | Correct denominator unsimplified | |
| \(= 27/55\) AG | B1 [3] | Answer given, so process needs to be convincing |
| OR \(3/12 \times 9/11 \times 8/10 \times {^3C_1}\) oe | M1 M1 | Mult 3 probs diff denoms (not \(a/3 \times b/4 \times c/5\)); Mult by \(^3C_1\) oe |
| \(= 27/55\) AG | A1 | Answer given, so process needs to be convincing |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(X\): 0, 1, 2, 3 | B1 | 0, 1, 2, 3 only seen in top line (condone additional values if Prob stated as 0) |
| Prob: \(84/220\), \(27/55\), \(27/220\), \(1/220\) | B1 | One correct prob, correctly placed in table |
| B1 | One other correct prob, correctly placed in table | |
| B1\(\checkmark\) [4] | One other correct prob ft \(\Sigma p = 1\), 4 values in table |
## Question 2:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(1 \text{ T-shirt}) = \dfrac{^3C_1 \times ^9C_2}{^{12}C_3}$ | B1 | Correct numerator unsimplified |
| | B1 | Correct denominator unsimplified |
| $= 27/55$ AG | B1 [3] | Answer given, so process needs to be convincing |
| **OR** $3/12 \times 9/11 \times 8/10 \times {^3C_1}$ oe | M1 M1 | Mult 3 probs diff denoms (not $a/3 \times b/4 \times c/5$); Mult by $^3C_1$ oe |
| $= 27/55$ AG | A1 | Answer given, so process needs to be convincing |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $X$: 0, 1, 2, 3 | B1 | 0, 1, 2, 3 only seen in top line (condone additional values if Prob stated as 0) |
| Prob: $84/220$, $27/55$, $27/220$, $1/220$ | B1 | One correct prob, correctly placed in table |
| | B1 | One other correct prob, correctly placed in table |
| | B1$\checkmark$ [4] | One other correct prob ft $\Sigma p = 1$, 4 values in table |
---2 Noor has 3 T-shirts, 4 blouses and 5 jumpers. She chooses 3 items at random. The random variable $X$ is the number of T-shirts chosen.\\
(i) Show that the probability that Noor chooses exactly one T-shirt is $\frac { 27 } { 55 }$.\\
(ii) Draw up the probability distribution table for $X$.
\hfill \mbox{\textit{CAIE S1 2016 Q2 [7]}}