| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2008 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Finding minimum n for P(X≥k) threshold |
| Difficulty | Standard +0.3 This is a straightforward Poisson distribution question requiring rate adjustment across different time periods. Part (i) and (ii) involve direct application of the Poisson formula with simple rate scaling. Part (iii) requires solving P(X≥1)=0.9, which reduces to 1-e^(-λt)=0.9, a simple logarithm manipulation. All steps are standard textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | \(\lambda = 0.8\) | M1 |
| \(P(2) = e^{-0.8} \frac{0.8^2}{2} = 0.144\) | A1 [2] | Correct answer |
| (ii) | \(\lambda = 0.2\) | M1 |
| \(P(0) = e^{-0.2} = 0.819\) | A1 [2] | Correct answer |
| (iii) | \(1 - e^{-0.8t} = 0.9\) | M1 |
| ln 0.1 = -0.8t | A1 | Correct equation with 0.8t |
| M1 | Attempt to solve equation by ln | |
| \(t = 2.88\) | A1 [4] | Correct answer |
(i) | $\lambda = 0.8$ | M1 | Attempt at Poisson calculation with attempt at $\lambda$
$P(2) = e^{-0.8} \frac{0.8^2}{2} = 0.144$ | A1 [2] | Correct answer
(ii) | $\lambda = 0.2$ | M1 | Attempt at P(0) with attempt at $\lambda$
$P(0) = e^{-0.2} = 0.819$ | A1 [2] | Correct answer
(iii) | $1 - e^{-0.8t} = 0.9$ | M1 | Equation containing at least 0.9 and $e^{-k}$
| ln 0.1 = -0.8t | A1 | Correct equation with 0.8t
| | M1 | Attempt to solve equation by ln
| $t = 2.88$ | A1 [4] | Correct answer
---6 People arrive randomly and independently at the elevator in a block of flats at an average rate of 4 people every 5 minutes.\\
(i) Find the probability that exactly two people arrive in a 1-minute period.\\
(ii) Find the probability that nobody arrives in a 15 -second period.\\
(iii) The probability that at least one person arrives in the next $t$ minutes is 0.9 . Find the value of $t$.
\hfill \mbox{\textit{CAIE S2 2008 Q6 [8]}}