| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2015 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | First success on specific trial |
| Difficulty | Easy -1.2 This is a straightforward application of basic probability and geometric distribution. Part (i) is trivial (1/4), part (ii) requires one application of the geometric distribution formula with clearly stated parameters, and part (iii) involves a simple counting argument (4/4 × 3/4 × 2/4 × 1/4). All parts are direct recall with minimal problem-solving, making this easier than average. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.02f Geometric distribution: conditions5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{1}{4}\) | B1 (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\left(\frac{3}{4}\right)^4\left(\frac{1}{4}\right) = \frac{81}{1024} = 0.0791\) | M1 | Expression of form \(p^4(1-p)\) only, \(p = 1/4\) or \(3/4\) |
| A1 (2) | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(\text{all diff}) = \frac{1}{4} \times \frac{1}{4} \times \frac{1}{4} \times \frac{1}{4} \times 4!\) | M1 | 4! on numerator seen mult by \(k \geq 1\) or \(3\times2\times1\) on num oe, must be in a fraction |
| \(= \frac{3}{32}\ (0.0938)\) | M1 | \(4^4\) on denom or \(4^3\) on denom with the \(3\times2\times1\) |
| OR \(1 \times \frac{3}{4} \times \frac{2}{4} \times \frac{1}{4} = \frac{3}{32}\) | A1 (3) | Correct answer |
## Question 3:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{1}{4}$ | B1 (1) | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\left(\frac{3}{4}\right)^4\left(\frac{1}{4}\right) = \frac{81}{1024} = 0.0791$ | M1 | Expression of form $p^4(1-p)$ only, $p = 1/4$ or $3/4$ |
| | A1 (2) | Correct answer |
### Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(\text{all diff}) = \frac{1}{4} \times \frac{1}{4} \times \frac{1}{4} \times \frac{1}{4} \times 4!$ | M1 | 4! on numerator seen mult by $k \geq 1$ or $3\times2\times1$ on num oe, must be in a fraction |
| $= \frac{3}{32}\ (0.0938)$ | M1 | $4^4$ on denom or $4^3$ on denom with the $3\times2\times1$ |
| OR $1 \times \frac{3}{4} \times \frac{2}{4} \times \frac{1}{4} = \frac{3}{32}$ | A1 (3) | Correct answer |
---3 One plastic robot is given away free inside each packet of a certain brand of biscuits. There are four colours of plastic robot (red, yellow, blue and green) and each colour is equally likely to occur. Nick buys some packets of these biscuits. Find the probability that\\
(i) he gets a green robot on opening his first packet,\\
(ii) he gets his first green robot on opening his fifth packet.
Nick's friend Amos is also collecting robots.\\
(iii) Find the probability that the first four packets Amos opens all contain different coloured robots.
\hfill \mbox{\textit{CAIE S1 2015 Q3 [6]}}