| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2006 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | At least one success |
| Difficulty | Moderate -0.3 This is a straightforward binomial distribution question testing standard techniques: (i) uses cumulative tables directly, (ii) applies the binomial probability formula, and (iii) requires the complement rule P(Z≥1)=1-P(Z=0) and solving an inequality. All parts are routine applications with no conceptual challenges, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| 4(i) | \(0.4207\) or \(0.421\) (3 sfs) or \(0.8^{–25}×0.8^{–n}×0.2 + \ldots + ^{\text{C}}_{x}0.4^{x}x0.2^r\) \(0.579(3)\) | B1: —; B1: 2 |
| 4(ii) | \(^{10}\text{C}_x(1-0.27)^x \times 0.27^y\) \(= 0.261\) (3 sfs) | M1: —; A1: 2 |
| 4(iii) | Allow "=" thro'out \(1 - 0.73^y > 0.95\) or \(0.73^y < 0.05\) | M1: —; M1: — |
| \(0.73^9 = 0.059\) \(0.73^{10} = 0.043\) | \(\text{nlog}0.73 < \text{log}0.05\) oe | |
| \(\text{ft}(1 - 0.27)\) from (ii) for M1M1 10 with incorrect sign in wking: SCB2 10 with just \(0.73^9 = 0.059\): M1M1A1 | ||
| \(n = 10\) | A1: 3 |
4(i) | $0.4207$ or $0.421$ (3 sfs) or $0.8^{–25}×0.8^{–n}×0.2 + \ldots + ^{\text{C}}_{x}0.4^{x}x0.2^r$ $0.579(3)$ | B1: —; B1: 2 | or $1 - 0.6167$ or $0.3833$ (3 sfs) or $1 - (6 \text{ correct terms}, 0 \text{ to } 5)$ |
4(ii) | $^{10}\text{C}_x(1-0.27)^x \times 0.27^y$ $= 0.261$ (3 sfs) | M1: —; A1: 2 | |
4(iii) | Allow "=" thro'out $1 - 0.73^y > 0.95$ or $0.73^y < 0.05$ | M1: —; M1: — | or $1 - ^{\text{C}}_0 \times 0.27^y × 0.73^y > 0.95$ oe allow incorrect sign M1 must be correct |
| $0.73^9 = 0.059$ $0.73^{10} = 0.043$ | | $\text{nlog}0.73 < \text{log}0.05$ oe | |
| | | $\text{ft}(1 - 0.27)$ from (ii) for M1M1 10 with incorrect sign in wking: SCB2 10 with just $0.73^9 = 0.059$: M1M1A1 | |
| $n = 10$ | A1: 3 | |
**Total: 7**
---4 (i) The random variable $X$ has the distribution $\mathrm { B } ( 25,0.2 )$. Using the tables of cumulative binomial probabilities, or otherwise, find $\mathrm { P } ( X \geqslant 5 )$.\\
(ii) The random variable $Y$ has the distribution $\mathrm { B } ( 10,0.27 )$. Find $\mathrm { P } ( Y = 3 )$.\\
(iii) The random variable $Z$ has the distribution $\mathrm { B } ( n , 0.27 )$. Find the smallest value of $n$ such that $\mathrm { P } ( Z \geqslant 1 ) > 0.95$.
\hfill \mbox{\textit{OCR S1 2006 Q4 [7]}}