| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2012 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Independent binomial samples with compound probability |
| Difficulty | Standard +0.8 This is a compound binomial problem requiring two nested applications: first calculating P(at least 1 day) from a binomial with n=7, p=0.21, then using this as the probability parameter for a second binomial with n=4. Part (i) is straightforward cumulative binomial calculation, but part (ii) requires recognizing the nested structure and careful probability manipulation, which elevates this above routine exercises. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(P(X < 5) = 1 - P(5, 6, 7)\) | M1 | Binomial expression with powers \(\sum\) 7 and probs \(\sum = 1\), and \(^nC_r\) |
| \(= 1 - (0.21)^5(0.79)^2 \cdot ^7C_5 - (0.21)^6(0.79)^1 \cdot ^7C_6 - (0.21)^7(0.79)^0 \cdot ^7C_7\) | ||
| \(= (0.21)^7\) | ||
| \(= 0.994\) | A1 [3] | Correct unsimplified expression |
| A1 | Correct answer | |
| (ii) \(P(\text{at least } 1) = 1 - P(0) = 1 - (0.79)^7\) | M1 | Attempt to find \(P(\text{at least } 1)\) or \(1 - P(0 \text{ and } 1)\) |
| \(= 0.808\) | A1 | Rounding to correct interval |
| \(P(\text{exactly } 3 \text{ weeks}) = (0.808)^3(0.192)_1C_3\) | M1 | Bin expression with powers \(\geq\) 4 and their 0.808 etc. and \(_nC_3\) |
| \(= 0.405\) | A1 [4] | Correct answer |
**(i)** $P(X < 5) = 1 - P(5, 6, 7)$ | M1 | Binomial expression with powers $\sum$ 7 and probs $\sum = 1$, and $^nC_r$
$= 1 - (0.21)^5(0.79)^2 \cdot ^7C_5 - (0.21)^6(0.79)^1 \cdot ^7C_6 - (0.21)^7(0.79)^0 \cdot ^7C_7$ |
$= (0.21)^7$ |
$= 0.994$ | A1 [3] | Correct unsimplified expression
| A1 | Correct answer
**(ii)** $P(\text{at least } 1) = 1 - P(0) = 1 - (0.79)^7$ | M1 | Attempt to find $P(\text{at least } 1)$ or $1 - P(0 \text{ and } 1)$
$= 0.808$ | A1 | Rounding to correct interval
$P(\text{exactly } 3 \text{ weeks}) = (0.808)^3(0.192)_1C_3$ | M1 | Bin expression with powers $\geq$ 4 and their 0.808 etc. and $_nC_3$
$= 0.405$ | A1 [4] | Correct answer4 In a certain mountainous region in winter, the probability of more than 20 cm of snow falling on any particular day is 0.21 .\\
(i) Find the probability that, in any 7-day period in winter, fewer than 5 days have more than 20 cm of snow falling.\\
(ii) For 4 randomly chosen 7-day periods in winter, find the probability that exactly 3 of these periods will have at least 1 day with more than 20 cm of snow falling.
\hfill \mbox{\textit{CAIE S1 2012 Q4 [7]}}