| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2014 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | State general binomial conditions |
| Difficulty | Moderate -0.8 Part (i) is pure recall of standard binomial conditions. Part (ii) is a straightforward binomial probability calculation using complement rule (1 - P(X ≤ 2)) with clearly stated parameters n=18, p=0.15. Both parts require only routine application of memorized facts and standard techniques with no problem-solving insight needed. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| (i) constant / given p, independent trials, fixed / given no. of trials, only two outcomes | B1 | Any one correct |
| B1 | 2 | Any 3 correct |
| (ii) \(P(x \geq 3) = 1 - P(0, 1, 2)\) | M1 | Any binomial expression \(p^r(1-p)^{18-r}\) \(^{18}C_r\) seen |
| \(= 1 - [(0.85)^{18} + (0.85)^{17}(0.15) \times 18 + (0.85)^{16}(0.15)^2 \times ^{18}C_2]\) | M1 | \(1 - P(0, 1, 2)\), any n,p,q |
| \(= 0.520\) | A1 | 3 |
(i) constant / given p, independent trials, fixed / given no. of trials, only two outcomes | B1 | Any one correct
| B1 | 2 | Any 3 correct
(ii) $P(x \geq 3) = 1 - P(0, 1, 2)$ | M1 | Any binomial expression $p^r(1-p)^{18-r}$ $^{18}C_r$ seen
$= 1 - [(0.85)^{18} + (0.85)^{17}(0.15) \times 18 + (0.85)^{16}(0.15)^2 \times ^{18}C_2]$ | M1 | $1 - P(0, 1, 2)$, any n,p,q
$= 0.520$ | A1 | 3 | Correct answer3 (i) State three conditions which must be satisfied for a situation to be modelled by a binomial distribution.
George wants to invest some of his monthly salary. He invests a certain amount of this every month for 18 months. For each month there is a probability of 0.25 that he will buy shares in a large company, there is a probability of 0.15 that he will buy shares in a small company and there is a probability of 0.6 that he will invest in a savings account.\\
(ii) Find the probability that George will buy shares in a small company in at least 3 of these 18 months.
\hfill \mbox{\textit{CAIE S1 2014 Q3 [5]}}