| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2012 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Combined probability with other distributions |
| Difficulty | Standard +0.3 This is a straightforward binomial probability question with n=20, p=0.05. Part (i) requires calculating P(X>1) = 1-P(X≤1), which is standard bookwork. Part (ii) adds a simple expected value calculation combining the binomial probability with basic arithmetic (revenue minus costs in two scenarios). While it requires multiple steps, each step is routine and the question clearly signposts what to do. Slightly above average difficulty due to the two-part structure and expected value context, but still a standard S1 exam question. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(>1) = 1 - (0.95)^{20} - (0.95)^{19}(0.05)^1 {_{20}}C_1\) | M1 | Binomial term \(20Cx(0.05)^x(0.95)^{20-x}\) |
| M1 | Correct unsimplified expression | |
| \(= 0.264\) | A1 [3] | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Profit 19 or 20 work \(= 450 \times 10 - 480 = 4020\) | B1 | 4020 seen |
| Profit \(< 19\) work \(= -480\) | M1 | Multiplying 4020 by their (i) or their \((1-\text{(i)})\) |
| Expected profit \(= \frac{4020 \times (1-0.264) - 480 \times 0.264}{1}\) | M1 | Multiplying 480 by \([1 - \text{their (i)}]\) and subtracting |
| \(= \\)2830\ (\\(2832)\) | A1 [4] | Rounding to correct answer |
| OR \(-480 + 4500(1-0.264) = 2830\) |
## Question 5:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(>1) = 1 - (0.95)^{20} - (0.95)^{19}(0.05)^1 {_{20}}C_1$ | M1 | Binomial term $20Cx(0.05)^x(0.95)^{20-x}$ |
| | M1 | Correct unsimplified expression |
| $= 0.264$ | A1 [3] | Correct answer |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Profit 19 or 20 work $= 450 \times 10 - 480 = 4020$ | B1 | 4020 seen |
| Profit $< 19$ work $= -480$ | M1 | Multiplying 4020 by their (i) or their $(1-\text{(i)})$ |
| Expected profit $= \frac{4020 \times (1-0.264) - 480 \times 0.264}{1}$ | M1 | Multiplying 480 by $[1 - \text{their (i)}]$ and subtracting |
| $= \$2830\ (\$2832)$ | A1 [4] | Rounding to correct answer |
| OR $-480 + 4500(1-0.264) = 2830$ | | |
---5 A company set up a display consisting of 20 fireworks. For each firework, the probability that it fails to work is 0.05 , independently of other fireworks.\\
(i) Find the probability that more than 1 firework fails to work.
The 20 fireworks cost the company $\$ 24$ each. 450 people pay the company $\$ 10$ each to watch the display. If more than 1 firework fails to work they get their money back.\\
(ii) Calculate the expected profit for the company.
\hfill \mbox{\textit{CAIE S1 2012 Q5 [7]}}