| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2015 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | Geometric with multiple success milestones |
| Difficulty | Standard +0.3 This is a straightforward application of geometric and binomial distributions with clear structure. Part (i) tests basic geometric distribution formulas, part (ii) is standard binomial probability, and part (iii) requires combining concepts but follows directly from definitions. The multi-part nature and need to recognize which distribution applies in part (iii) elevates it slightly above average, but all techniques are routine for S1 students. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities5.02f Geometric distribution: conditions5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1) |
| Answer | Marks | Guidance |
|---|---|---|
| \((1 - 0.27)^7 \times 0.27\) | M1 | alone |
| \(= 0.0298\) (3 sf) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \((1 - 0.27)^8\) | M1 | alone; or \(1 - P(X = 1,2,3,4,5,6,7,8)\) all terms correct \((= 1 - 0.91935)\); NOT \((1-0.27)^8 \times \ldots\); NOT \(1-(1-0.27)^8\) |
| \(= 0.0806\) (3 sf) or \(0.08065\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Bin stated | B1 | or implied by \(^8C_2\) or \(^8C_6\) or \((1-0.27)^a \times 0.27^b\ (a+b=8)\); by ans 0.309; Allow "Bio"; Allow correct \(+\ldots\); Correct ans, no working: B1B0B1 |
| \(^8C_2 \times (1-0.27)^6 \times 0.27^2\) | B1 | NOTE: Must see sub in formula for this B1 |
| \(0.309\) (3 sf) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Their (ii) \(\times 0.27\) seen together | M1 | or \((^8C_2\times(1-0.27)^6\times0.27^2)\times 0.27\) seen together; or \(^8C_2\times(1-0.27)^8\times0.27^4\) SC: \((1-0.27)^8\times0.27^4\) alone M0M1A0 |
| Their (ii) \(\times 0.27 \times (1-0.27)^2 \times 0.27\) ie wholly correct method ft(ii) | M1 | or \(^8C_2\times(1-0.27)^6\times0.27^2\times0.27\times(1-0.27)^2\times0.27\) ie wholly correct method |
| \(= 0.0120\) (3 sf) | A1ft | Allow 0.012; ft their (ii) only |
# Question 5:
## Part (i)(a)
$(1 - 0.27)^7 \times 0.27$ | M1 | alone
$= 0.0298$ (3 sf) | A1 |
**[2]**
## Part (i)(b)
$(1 - 0.27)^8$ | M1 | alone; or $1 - P(X = 1,2,3,4,5,6,7,8)$ all terms correct $(= 1 - 0.91935)$; NOT $(1-0.27)^8 \times \ldots$; NOT $1-(1-0.27)^8$
$= 0.0806$ (3 sf) or $0.08065$ | A1 |
**[2]**
## Part (ii)
Bin stated | B1 | or implied by $^8C_2$ or $^8C_6$ or $(1-0.27)^a \times 0.27^b\ (a+b=8)$; by ans 0.309; Allow "Bio"; Allow correct $+\ldots$; Correct ans, no working: B1B0B1
$^8C_2 \times (1-0.27)^6 \times 0.27^2$ | B1 | NOTE: Must see sub in formula for this B1
$0.309$ (3 sf) | B1 |
**[3]**
## Part (iii)
Their (ii) $\times 0.27$ seen together | M1 | or $(^8C_2\times(1-0.27)^6\times0.27^2)\times 0.27$ seen together; or $^8C_2\times(1-0.27)^8\times0.27^4$ SC: $(1-0.27)^8\times0.27^4$ alone M0M1A0
Their (ii) $\times 0.27 \times (1-0.27)^2 \times 0.27$ ie wholly correct method ft(ii) | M1 | or $^8C_2\times(1-0.27)^6\times0.27^2\times0.27\times(1-0.27)^2\times0.27$ ie wholly correct method
$= 0.0120$ (3 sf) | A1ft | Allow 0.012; ft their (ii) only
**[3]**
---5 Each year Jack enters a ballot for a concert ticket. The probability that Jack will win a ticket in any particular year is 0.27 .\\
(i) Find the probability that the first time Jack wins a ticket is
\begin{enumerate}[label=(\alph*)]
\item on his 8th attempt,
\item after his 8th attempt.\\
(ii) Write down an expression for the probability that Jack wins a ticket on exactly 2 of his first 8 attempts, and evaluate this expression.\\
(iii) Find the probability that Jack wins his 3rd ticket on his 9th attempt and his 4th ticket on his 12th attempt.
\end{enumerate}
\hfill \mbox{\textit{OCR S1 2015 Q5 [10]}}