| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2007 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Verify conditions in context |
| Difficulty | Moderate -0.3 This is a straightforward binomial distribution question requiring recall of conditions (part i), direct use of cumulative probability tables (part ii), and simple complement/conditional probability calculations (parts iii-iv). All parts follow standard S1 patterns with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Packets indep oe | B1 | Context essential |
| B1 | 2 | NOT vouchers indep |
| Answer | Marks | Guidance |
|---|---|---|
| 0.9857 or 0.986 (3 sfs) | B2 | 2 |
| or for 7 terms correct, allow one omit or | ||
| extra | ||
| NOT \(1 - 0.9857 = 0.0143\) (see (iii)) |
| Answer | Marks | Guidance |
|---|---|---|
| \((1 - 0.9857) = 0.014(3)\) (2 sfs) | B1ft | Allow 1- their (ii) correctly calc'd |
| 1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(6 \text{ from } 11) × 0.25 = 0.00669\) or \(6.69 × 10^{-3}\) (3 sfs) | B1 | or \(0.75^5 × 0.25^6\) (\(a + b = 11\)) or \(^{11}C_6\) |
| M1 | dep B1 | |
| M1 | ||
| A1 | 4 |
### Part i
$P($contains voucher$)$ constant oe
Packets indep oe | B1 | Context essential
| B1 | 2 | NOT vouchers indep
### Part ii
0.9857 or 0.986 (3 sfs) | B2 | 2 | B1 for 0.9456 or 0.946 or 0.997(2)
| | or for 7 terms correct, allow one omit or
| | extra
| | NOT $1 - 0.9857 = 0.0143$ (see (iii))
### Part iii
$(1 - 0.9857) = 0.014(3)$ (2 sfs) | B1ft | Allow 1- their (ii) correctly calc'd
| 1 |
### Part iv
$B(11, 0.25)$ or 6 in 11 wks stated or impl
$^{11}C_6 × 0.75^5 × 0.25^6$ (= 0.0267663)
$P(6 \text{ from } 11) × 0.25 = 0.00669$ or $6.69 × 10^{-3}$ (3 sfs) | B1 | or $0.75^5 × 0.25^6$ ($a + b = 11$) or $^{11}C_6$
| M1 | dep B1
| M1 |
| A1 | 4
**Total: 9**
---7 On average, $25 \%$ of the packets of a certain kind of soup contain a voucher. Kim buys one packet of soup each week for 12 weeks. The number of vouchers she obtains is denoted by X .\\
(i) State two conditions needed for X to be modelled by the distribution $\mathrm { B } ( 12,0.25 )$.
In the rest of this question you should assume that these conditions are satisfied.\\
(ii) Find $\mathrm { P } ( \mathrm { X } \leqslant 6 )$.
In order to claim a free gift, 7 vouchers are needed.\\
(iii) Find the probability that Kim will be able to claim a free gift at some time during the 12 weeks.\\
(iv) Find the probability that Kim will be able to claim a free gift in the 12th week but not before.
\hfill \mbox{\textit{OCR S1 2007 Q7 [9]}}