| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2021 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Geometric distribution (first success) |
| Difficulty | Moderate -0.8 This question involves straightforward applications of binomial and geometric distributions with clear parameters. Part (a) is a routine binomial calculation, part (b) applies the geometric distribution formula directly, and part (c) combines the result from (b) with another simple binomial calculation. All parts require only standard formula application with no problem-solving insight or complex multi-step reasoning. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \([P(0,1,2) =]\ {}^{10}C_0\, 0.16^0\, 0.84^{10} + {}^{10}C_1\, 0.16^1\, 0.84^9 + {}^{10}C_2\, 0.16^2\, 0.84^8\) \([= 0.17490 + 0.333145 + 0.28555]\) | M1 | One term: \({}^{10}C_x\, p^x\,(1-p)^{10-x}\) for \(0 \leq x < 10\), any \(p\) |
| A1 | Correct unsimplified expression, or better | |
| 0.794 | A1 | \(0.7935 < p \leqslant 0.794\), mark at most accurate. If M0 scored, SC B1 for final answer 0.794 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((0.84)^7\, 0.16\) | M1 | \((1-p)^7 p\), \(0 < p < 1\) |
| 0.0472 | A1 | 0.0472144 to at least 3sf |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(4 \times 0.0472 \times (1 - 0.0472)^3\) | M1 | \(4 \times q(1-q)^3\), \(q =\) *their* (b) or correct |
| 0.163 | A1 | \(0.163 \leqslant p \leqslant 0.1634\), mark at most accurate from *their* probability to at least 3sf |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $[P(0,1,2) =]\ {}^{10}C_0\, 0.16^0\, 0.84^{10} + {}^{10}C_1\, 0.16^1\, 0.84^9 + {}^{10}C_2\, 0.16^2\, 0.84^8$ $[= 0.17490 + 0.333145 + 0.28555]$ | M1 | One term: ${}^{10}C_x\, p^x\,(1-p)^{10-x}$ for $0 \leq x < 10$, any $p$ |
| | A1 | Correct unsimplified expression, or better |
| 0.794 | A1 | $0.7935 < p \leqslant 0.794$, mark at most accurate. If M0 scored, SC B1 for final answer 0.794 |
---
## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $(0.84)^7\, 0.16$ | M1 | $(1-p)^7 p$, $0 < p < 1$ |
| 0.0472 | A1 | 0.0472144 to at least 3sf |
---
## Question 5(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $4 \times 0.0472 \times (1 - 0.0472)^3$ | M1 | $4 \times q(1-q)^3$, $q =$ *their* **(b)** or correct |
| 0.163 | A1 | $0.163 \leqslant p \leqslant 0.1634$, mark at most accurate from *their* probability to at least 3sf |
---5 In a certain region, the probability that any given day in October is wet is 0.16 , independently of other days.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that, in a 10-day period in October, fewer than 3 days will be wet.
\item Find the probability that the first wet day in October is 8 October.
\item For 4 randomly chosen years, find the probability that in exactly 1 of these years the first wet day in October is 8 October.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2021 Q5 [7]}}