| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2005 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | Mean/expectation of geometric distribution |
| Difficulty | Moderate -0.8 This is a straightforward application of the geometric distribution formula with direct recall of standard results. Part (i) requires plugging values into P(T=k) = (1-p)^(k-1)p and summing a geometric series, while part (ii) simply asks to state the mean formula E(T)=1/p. No problem-solving or conceptual insight needed beyond recognizing the distribution and applying memorized formulas. |
| Spec | 5.02g 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 |
|---|---|---|
| (i) (a) \(\text{Geo}(0.14)\) stated in (a) or (b) | B1 | or \(0.86^n \times 0.14\) or \(0.14^n \times 0.86\) in (a) or \(\text{M1}\) in (b) |
| \((0.86)^3 \times 0.14 = 0.0766\) (3 sfs) | M1 A1 | 3 |
| (b) \(1 - 0.86\) or \(0.14 + 0.86 \times 0.14 + ... + 0.86^n \times 0.14 = 0.652\) (3 sfs) | M2 A1 | 3 |
| (ii) \(\frac{1/0.14} = 7\frac{1}{7}\) or \(7.14\) (3 sfs) | M1 A1 | 2 |
(i) (a) $\text{Geo}(0.14)$ stated in (a) or (b) | B1 | or $0.86^n \times 0.14$ or $0.14^n \times 0.86$ in (a) or $\text{M1}$ in (b)
$(0.86)^3 \times 0.14 = 0.0766$ (3 sfs) | M1 A1 | 3 | No working: $0.077$; B1M1A0
(b) $1 - 0.86$ or $0.14 + 0.86 \times 0.14 + ... + 0.86^n \times 0.14 = 0.652$ (3 sfs) | M2 A1 | 3 | $1 - 0.86 + 8^{\text{th}}$ term ($r = 7$ or 0) or 1 missing term: M1
(ii) $\frac{1/0.14} = 7\frac{1}{7}$ or $7.14$ (3 sfs) | M1 A1 | 2 |2 The probability that a certain sample of radioactive material emits an alpha-particle in one unit of time is 0.14 . In one unit of time no more than one alpha-particle can be emitted. The number of units of time up to and including the first in which an alpha-particle is emitted is denoted by $T$.\\
(i) Find the value of
\begin{enumerate}[label=(\alph*)]
\item $\mathrm { P } ( T = 5 )$,
\item $\mathrm { P } ( T < 8 )$.\\
(ii) State the value of $\mathrm { E } ( T )$.
\end{enumerate}
\hfill \mbox{\textit{OCR S1 2005 Q2 [8]}}