| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2007 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Direct binomial probability calculation |
| Difficulty | Moderate -0.8 This is a straightforward application of binomial distribution formulas and properties. Part (i) is direct substitution into the binomial probability formula, part (ii) is a standard normal approximation with continuity correction, and part (iii) requires only solving np ≥ 8. All three parts are routine textbook exercises requiring recall of standard techniques with no problem-solving insight or multi-step reasoning. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04d Normal approximation to binomial |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(X=5) = (0.65)^5 \times (0.35)^2 \times {}_7C_5\) | M1 | Expression with 3 terms, powers summing to 7 and a \(_7C\) term |
| \(= 0.298\), allow \(0.2985\) | A1 [2] | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\mu = 50 \times 0.65 = 32.5\); \(\sigma^2 = 50 \times 0.65 \times 0.35 = 11.375\) | B1 | \(32.5\) and \(11.375\) seen or implied |
| \(P(\text{fewer than } 29) = \Phi\!\left(\frac{28.5 - 32.5}{\sqrt{11.375}}\right)\) | M1 | Standardising, with or without cc, must have sq rt |
| M1 | For continuity correction \(28.5\) or \(29.5\) | |
| \(= 1 - \Phi(1.186)\) | M1 | Correct area ie \(< 0.5\) must be from a normal approx |
| \(= 1 - 0.8822 = 0.118\) | A1 [5] | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(0.65\,n \geq 8\) | M1 | Equality or inequality with \(np\) and 8 |
| Smallest \(n = 13\) | A1 [2] | Correct answer |
## Question 6:
**Part (i)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(X=5) = (0.65)^5 \times (0.35)^2 \times {}_7C_5$ | M1 | Expression with 3 terms, powers summing to 7 and a $_7C$ term |
| $= 0.298$, allow $0.2985$ | A1 **[2]** | Correct answer |
**Part (ii)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mu = 50 \times 0.65 = 32.5$; $\sigma^2 = 50 \times 0.65 \times 0.35 = 11.375$ | B1 | $32.5$ and $11.375$ seen or implied |
| $P(\text{fewer than } 29) = \Phi\!\left(\frac{28.5 - 32.5}{\sqrt{11.375}}\right)$ | M1 | Standardising, with or without cc, must have sq rt |
| | M1 | For continuity correction $28.5$ or $29.5$ |
| $= 1 - \Phi(1.186)$ | M1 | Correct area ie $< 0.5$ must be from a normal approx |
| $= 1 - 0.8822 = 0.118$ | A1 **[5]** | Correct answer |
**Part (iii)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $0.65\,n \geq 8$ | M1 | Equality or inequality with $np$ and 8 |
| Smallest $n = 13$ | A1 **[2]** | Correct answer |
---6 On any occasion when a particular gymnast performs a certain routine, the probability that she will perform it correctly is 0.65 , independently of all other occasions.\\
(i) Find the probability that she will perform the routine correctly on exactly 5 occasions out 7 .\\
(ii) On one day she performs the routine 50 times. Use a suitable approximation to estimate the probability that she will perform the routine correctly on fewer than 29 occasions.\\
(iii) On another day she performs the routine $n$ times. Find the smallest value of $n$ for which the expected number of correct performances is at least 8 .
\hfill \mbox{\textit{CAIE S1 2007 Q6 [9]}}