| Exam Board | OCR |
|---|---|
| Module | FS1 AS (Further Statistics 1 AS) |
| Year | 2018 |
| Session | March |
| Marks | 7 |
| Topic | Geometric Distribution |
| Type | State assumptions for geometric model |
| Difficulty | Easy -1.2 This is a straightforward recall question testing basic knowledge of the geometric distribution. Part (i) requires stating standard assumptions (constant probability, independence), while parts (ii)-(iv) involve direct application of standard formulas with no problem-solving or conceptual challenge. This is easier than average A-level content. |
| Spec | 5.02f Geometric distribution: conditions5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1)5.02h Geometric: mean 1/p and variance (1-p)/p^2 |
| Answer | Marks | Guidance |
|---|---|---|
| Outcome of each attempt is independent of the outcome of any other attempt | B1 | Must be contextualised for any marks - NOT "probability of passing at all is independent" |
| Probability of passing on any attempt is the same | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 - 0.6^7\) or \(1 - 0.6^6\) | M1 | Allow \(1 - 0.6^7\) or \(1 - 0.6^6\) |
| \(= 0.92224\) | A1 | Answer, awrt 0.922; Or \(p\sum_{r=0}^{6} q^r\) allow \(\pm 1\) term for M1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(X) = 1/p = 2\frac{1}{2}\) | B1 | Any exact equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| \(\text{Var}(X) = q/p^2 = 0.6/0.4^2\) | M1 | Accept 0.6÷0.16 |
| \(3\frac{3}{4}\) | A1 | Any exact equivalent |
### Part (i)
Outcome of each attempt is independent of the outcome of any other attempt | **B1** | Must be contextualised for any marks - NOT "probability of passing at all is independent"
Probability of passing on any attempt is the same | **B1** |
### Part (ii)
$1 - 0.6^7$ or $1 - 0.6^6$ | **M1** | Allow $1 - 0.6^7$ or $1 - 0.6^6$
$= 0.92224$ | **A1** | Answer, awrt 0.922; Or $p\sum_{r=0}^{6} q^r$ allow $\pm 1$ term for M1
### Part (iii)
$E(X) = 1/p = 2\frac{1}{2}$ | **B1** | Any exact equivalent
### Part (iv)
$\text{Var}(X) = q/p^2 = 0.6/0.4^2$ | **M1** | Accept 0.6÷0.16
$3\frac{3}{4}$ | **A1** | Any exact equivalent
---1 A learner driver keeps taking the driving test until she passes. The number of attempts taken, up to and including the pass, is denoted by $X$.\\
(i) State two assumptions needed for $X$ to be well modelled by a geometric distribution.
Assume now that $X \sim \operatorname { Geo } ( 0.4 )$.\\
(ii) Find $\mathrm { P } ( X < 6 )$.\\
(iii) Find $\mathrm { E } ( X )$.\\
(iv) Find $\operatorname { Var } ( X )$.
\hfill \mbox{\textit{OCR FS1 AS 2018 Q1 [7]}}