| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2013 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Finding binomial parameters from properties |
| Difficulty | Standard +0.8 Part (i) is straightforward binomial probability calculation (p=4/9, n=5, P(X≥2)). Part (ii) requires solving simultaneous equations using np=96 and np(1-p)=32 to find both n and k, which demands algebraic manipulation and understanding that p=k/9. This is above-average difficulty due to the reverse-engineering of parameters from mean and variance, though the method is systematic once recognized. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04d Normal approximation to binomial5.01a Permutations and combinations: evaluate probabilities5.02d Binomial: mean np and variance np(1-p) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(p = 4/9\) or \(5/9\) | B1 | Binomial term \({}_{5}C_x p^x(1-p)^{5-x}\) seen |
| \(P(\text{at least } 2) = 1 - P(0,1)\) | M1 | |
| \(= 1 - (5/9)^5 - (4/9)(5/9)^4 \cdot {}_{5}C_1\) | ||
| \(= 0.735\) | A1 [3] | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(np = 96,\ npq = 32,\ p = P(\leq k)\) | M1 | Using \(np = 96\), \(npq = 32\) to obtain eqn in 1 variable |
| \(p = 2/3,\ q = 1/3,\ n = 144\) | A1 | \(1/3\) or \(2/3\) seen or implied |
| \(k = 6\) | A1ft | Correct \(k\) ft \(k = 9p\) |
| \(n = 144\) | A1 [4] | Correct \(n\) |
## Question 4:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $p = 4/9$ or $5/9$ | B1 | Binomial term ${}_{5}C_x p^x(1-p)^{5-x}$ seen |
| $P(\text{at least } 2) = 1 - P(0,1)$ | M1 | |
| $= 1 - (5/9)^5 - (4/9)(5/9)^4 \cdot {}_{5}C_1$ | | |
| $= 0.735$ | A1 **[3]** | Correct answer |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $np = 96,\ npq = 32,\ p = P(\leq k)$ | M1 | Using $np = 96$, $npq = 32$ to obtain eqn in 1 variable |
| $p = 2/3,\ q = 1/3,\ n = 144$ | A1 | $1/3$ or $2/3$ seen or implied |
| $k = 6$ | A1ft | Correct $k$ ft $k = 9p$ |
| $n = 144$ | A1 **[4]** | Correct $n$ |
---4 Robert uses his calculator to generate 5 random integers between 1 and 9 inclusive.\\
(i) Find the probability that at least 2 of the 5 integers are less than or equal to 4 .
Robert now generates $n$ random integers between 1 and 9 inclusive. The random variable $X$ is the number of these $n$ integers which are less than or equal to a certain integer $k$ between 1 and 9 inclusive. It is given that the mean of $X$ is 96 and the variance of $X$ is 32 .\\
(ii) Find the values of $n$ and $k$.
\hfill \mbox{\textit{CAIE S1 2013 Q4 [7]}}