| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2017 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Finding binomial parameters from properties |
| Difficulty | Moderate -0.8 This question tests basic binomial distribution properties using straightforward formulas. Part (i) requires simple division (E(X) = np → p = 6.21/30), part (ii) applies the variance formula directly (np(1-p)), and part (iii) is a standard binomial probability calculation using complement rule. All steps are routine applications of memorized formulas with no problem-solving or conceptual insight required. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(p = 0.207\) | B1 | |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{Var} = 30 \times 0.207 \times 0.793 = 4.92\) | B1 | |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(\geqslant 2) = 1 - P(0, 1)\) | M1 | |
| \(= 1 - (0.793)^{15} - \dbinom{15}{1}(0.207)(0.793)^{14}\) | M1 | \(1 - P(0,1)\) seen; \(n = 15\), \(p =\) any prob |
| \(= 0.848\) | A1 | |
| Total: 3 |
## Question 3(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $p = 0.207$ | B1 | |
| **Total: 1** | | |
---
## Question 3(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Var} = 30 \times 0.207 \times 0.793 = 4.92$ | B1 | |
| **Total: 1** | | |
---
## Question 3(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(\geqslant 2) = 1 - P(0, 1)$ | M1 | |
| $= 1 - (0.793)^{15} - \dbinom{15}{1}(0.207)(0.793)^{14}$ | M1 | $1 - P(0,1)$ seen; $n = 15$, $p =$ any prob |
| $= 0.848$ | A1 | |
| **Total: 3** | | |3 An experiment consists of throwing a biased die 30 times and noting the number of 4 s obtained. This experiment was repeated many times and the average number of 4 s obtained in 30 throws was found to be 6.21.\\
(i) Estimate the probability of throwing a 4.\\
..................................................................................................................................... .\\
\section*{Hence}
(ii) find the variance of the number of 4 s obtained in 30 throws,\\
(iii) find the probability that in 15 throws the number of 4 s obtained is 2 or more.\\
\hfill \mbox{\textit{CAIE S1 2017 Q3 [5]}}