| Exam Board | OCR |
|---|---|
| Module | FS1 AS (Further Statistics 1 AS) |
| Year | 2017 |
| Session | December |
| Marks | 8 |
| Topic | Binomial Distribution |
| Type | Geometric distribution (first success) |
| Difficulty | Moderate -0.3 This is a straightforward application of standard binomial and geometric distribution formulas. Part (i) requires direct recall of mean=np and variance=np(1-p). Part (ii)(a) asks for standard assumptions, and part (ii)(b) involves solving P(L≤n)≥0.9 using the geometric CDF formula, which is a routine textbook exercise requiring only algebraic manipulation of 1-(0.95)^n≥0.9. No novel insight or complex problem-solving is needed, making it slightly easier than average. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.02g 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 |
|---|---|---|
| \(\mu = np = 2\) | B1 | 2 only, allow 2.00 |
| \(\sigma^2 = npq = 1.9\) | B1 | Exact or awrt 1.90 |
| Answer | Marks |
|---|---|
| Responses are independent of one another | B1 |
| Each addressee is equally likely to respond favourably | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(0.95^n \leq 0.1\) | M1 | Equation, stated or implied; Allow = or reversed ≥; T&I: M1B2B1 |
| \(n \geq \log(0.1) \div \log(0.95)\) | M1 | or \(\log_{0.95}\) |
| \(n \geq 44.9\) | A1 | 44 from correct working: A1A0 |
| 45 letters | A1 | Allow "\(n_{min} = 45\)" |
## (i)
$\mu = np = 2$ | B1 | 2 only, allow 2.00
$\sigma^2 = npq = 1.9$ | B1 | Exact or awrt 1.90
## (ii)(a)
Responses are independent of one another | B1 |
Each addressee is equally likely to respond favourably | B1 |
## (ii)(b)
$0.95^n \leq 0.1$ | M1 | Equation, stated or implied; Allow = or reversed ≥; T&I: M1B2B1
$n \geq \log(0.1) \div \log(0.95)$ | M1 | or $\log_{0.95}$
$n \geq 44.9$ | A1 | 44 from correct working: A1A0
45 letters | A1 | Allow "$n_{min} = 45$"
---1 Bill and Gill send letters to potential sponsors of a show. On past experience, they know that $5 \%$ of letters receive a favourable reply.\\
(i) Bill sends a letter to each of 40 potential sponsors. Assuming that the number $N$ of favourable responses can be modelled by a binomial distribution, find the mean and variance of $N$.\\
(ii) Gill sends one letter at a time to potential sponsors. $L$ is the number of letters she sends, up to and including the first letter that receives a favourable response.
\begin{enumerate}[label=(\alph*)]
\item State two assumptions needed for $L$ to be well modelled by a geometric distribution.
\item Using the assumptions in part (ii)(a), find the smallest number of letters that Gill has to send in order to have at least a $90 \%$ chance of receiving at least one favourable reply.
\end{enumerate}
\hfill \mbox{\textit{OCR FS1 AS 2017 Q1 [8]}}