| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2018 |
| Session | June |
| Marks | 6 |
| Topic | Binomial Distribution |
| Type | General probability threshold |
| Difficulty | Moderate -0.3 This is a straightforward binomial distribution question requiring only standard calculations: (i) is trivial expectation (np), (ii) is direct cumulative probability P(X<14), and (iii) requires finding k by trial with cumulative probabilities. All parts use routine binomial formulas with no conceptual challenges or problem-solving insight needed, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
**Question 4(i)**
$16 \times 0.75 = 12$ **B1**
**Question 4(ii)**
$B(16, 0.75)$ seen or implied **B1** May be seen in **(i)**
Use tables to get $0.803$ **B1** (0.8029)
**Question 4(iii)**
$P(X < k) \leqslant 0.1$ **M1** May be implied; allow $P(X \leqslant k) \leqslant 0.1$
Any probability in list $(8, 0.0075)$, $(9, 0.0271)$, $(10, 0.0796)$, $(11, 0.1897)$ seen **A1**
(Pick $p = 0.0796$, hence) $k = 10$ **A1**4 On a particular day at a busy international airport, 75\% of the scheduled flights depart on time. A random sample of 16 flights is chosen.\\
(i) Find the expected number of flights that depart on time.\\
(ii) For these 16 flights, find the probability that fewer than 14 flights depart on time.\\
(iii) For these 16 flights, the probability that at least $k$ flights depart on time is greater than 0.9 . Find the largest possible value of $k$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2018 Q4 [6]}}