| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2014 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Direct binomial probability calculation |
| Difficulty | Easy -1.2 This is a straightforward application of binomial distribution requiring only direct substitution into the probability formula for part (i) and cumulative probability calculation for part (ii). Both parts involve routine use of given parameters with no problem-solving insight needed, making it easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\max = 12\), \(P(12) = (0.7)^{12} = 0.0138\) | B1 B1 2 | (Implied by P(12) with power 12) Accept 0.014 |
| (ii) \(P(\text{fewer than } 10) = 1 - P(10, 11, 12)\) | M1 | Binomial term \(^{12}C_r(0.7)^r(0.3)^{12-r}\) or \(^{12}C_r(p)^{12-r}\), 0.99 \(\le p + q \le\) 1.00 |
| \(= 1 - ^{12}C_{10} \times (0.7)^{10}(0.3)^2 - 12 \times (0.7)^{11}(0.3) - (0.7)^{12}\) | A1 | Correct unsimplified expression oe |
| \(= 1 - 0.2528 = 0.747\) | A1 3 | Correct answer |
**(i)** $\max = 12$, $P(12) = (0.7)^{12} = 0.0138$ | B1 B1 2 | (Implied by P(12) with power 12) Accept 0.014
**(ii)** $P(\text{fewer than } 10) = 1 - P(10, 11, 12)$ | M1 | Binomial term $^{12}C_r(0.7)^r(0.3)^{12-r}$ or $^{12}C_r(p)^{12-r}$, 0.99 $\le p + q \le$ 1.00
$= 1 - ^{12}C_{10} \times (0.7)^{10}(0.3)^2 - 12 \times (0.7)^{11}(0.3) - (0.7)^{12}$ | A1 | Correct unsimplified expression oe
$= 1 - 0.2528 = 0.747$ | A1 3 | Correct answer3 The number of books read by members of a book club each year has the binomial distribution $B ( 12,0.7 )$.\\
(i) State the greatest number of books that could be read by a member of the book club in a particular year and find the probability that a member reads this number of books.\\
(ii) Find the probability that a member reads fewer than 10 books in a particular year.
\hfill \mbox{\textit{CAIE S1 2014 Q3 [5]}}