| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2005 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Expected profit or cost problem |
| Difficulty | Standard +0.3 This is a straightforward expected value problem requiring calculation of a probability distribution using geometric probability (independent trials with constant probability). Part (i) is trivial arithmetic, part (ii) involves basic probability calculations with complementary events, and parts (iii)-(iv) are standard textbook exercises in constructing and using discrete probability distributions. The calculations are routine with no conceptual challenges beyond S1 level. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| For correct answer | B1 | 1 mark |
| Answer | Marks | Guidance |
|---|---|---|
| For attempting to sum \(P(\text{MMMH})\) and \(P(\text{MMMMH})\) | M1 | |
| For correct answer | A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(x\) | 4 | 2 |
| \(P(X = x)\) | 0.2 | 0.288 |
| For one correct prob other than 0.184 | B1 | |
| For another correct prob other than 0.184, ft only if the \(-1\) ignored and their 3rd prob is 1 – Σ the other 2 | B1 ft | |
| For correct table, can have separate 2s | B1 | 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| For attempt at \(\sum xp\) from their table, at least 2 non-zero terms | M1 | |
| For correct answer | A1 | 2 marks |
**(i)** $\$2$
| For correct answer | B1 | 1 mark
**(ii)** $P(\text{MMMH}) + P(\text{MMMMH}) = 0.8^3 \times 0.2 + 0.8^4 \times 0.2 = 0.184$ AG
| For attempting to sum $P(\text{MMMH})$ and $P(\text{MMMMH})$ | M1
| For correct answer | A1 | 2 marks
**(iii)**
| $x$ | 4 | 2 | 0 | -1 |
|---|---|---|---|---|
| $P(X = x)$ | 0.2 | 0.288 | 0.184 | 0.328 |
| For one correct prob other than 0.184 | B1
| For another correct prob other than 0.184, ft only if the $-1$ ignored and their 3rd prob is 1 – Σ the other 2 | B1 ft
| For correct table, can have separate 2s | B1 | 3 marks
**(iv)** $E(X) = 0.8 + 0.576 - 0.328 = \$1.05$
| For attempt at $\sum xp$ from their table, at least 2 non-zero terms | M1
| For correct answer | A1 | 2 marks6 In a competition, people pay $\$ 1$ to throw a ball at a target. If they hit the target on the first throw they receive $\$ 5$. If they hit it on the second or third throw they receive $\$ 3$, and if they hit it on the fourth or fifth throw they receive $\$ 1$. People stop throwing after the first hit, or after 5 throws if no hit is made. Mario has a constant probability of $\frac { 1 } { 5 }$ of hitting the target on any throw, independently of the results of other throws.\\
(i) Mario misses with his first and second throws and hits the target with his third throw. State how much profit he has made.\\
(ii) Show that the probability that Mario's profit is $\$ 0$ is 0.184 , correct to 3 significant figures.\\
(iii) Draw up a probability distribution table for Mario's profit.\\
(iv) Calculate his expected profit.
\hfill \mbox{\textit{CAIE S1 2005 Q6 [8]}}