| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating Binomial to Normal Distribution |
| Type | Exact binomial then normal approximation (same context, different n) |
| Difficulty | Standard +0.3 This is a straightforward application of Poisson distribution (parts a-b) and Poisson-to-Normal approximation (part c). The question requires standard probability calculations with clear parameters (λ=1 for one week, λ=7 for seven weeks) and routine use of continuity correction. While part (c) involves multiple steps, all techniques are textbook procedures with no conceptual challenges or novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04d Normal approximation to binomial5.02i Poisson distribution: random events model5.02k Calculate Poisson probabilities5.05a Sample mean distribution: central limit theorem |
| Answer | Marks |
|---|---|
| (a) \(X \sim B(5, 0.2)\), \(P(X = 0) = 0.8^5 = 0.3277\) | M1 A1 |
| (b) \(P(X > 2) = 1 - P(X \leq 2) = 1 - 0.9421 = 0.0579\) (from tables) | M1 A1 A1 |
| (c) No. of lates in 7 weeks is distributed \(B(35, 0.2) \approx N(7, 5.6)\) | M1 A1 |
| \(P(X > 10) = P(X > 10.5) = P\left(Z > \frac{3.5}{\sqrt{5.6}}\right) = P(Z > 1.48)\) | M1 A1 A1 |
| \(= 1 - 0.9306 = 0.0694\) | M1 A1 |
(a) $X \sim B(5, 0.2)$, $P(X = 0) = 0.8^5 = 0.3277$ | M1 A1 |
(b) $P(X > 2) = 1 - P(X \leq 2) = 1 - 0.9421 = 0.0579$ (from tables) | M1 A1 A1 |
(c) No. of lates in 7 weeks is distributed $B(35, 0.2) \approx N(7, 5.6)$ | M1 A1 |
$P(X > 10) = P(X > 10.5) = P\left(Z > \frac{3.5}{\sqrt{5.6}}\right) = P(Z > 1.48)$ | M1 A1 A1 |
$= 1 - 0.9306 = 0.0694$ | M1 A1 |
**Total: 12 marks**
---A certain Sixth Former is late for school once a week, on average. In a particular week of 5 days, find the probability that
\begin{enumerate}[label=(\alph*)]
\item he is not late at all, [2 marks]
\item he is late more than twice. [3 marks]
\end{enumerate}
In a half term of seven weeks, lateness on more than ten occasions results in loss of privileges the following half term.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Use the Normal approximation to estimate the probability that he loses his privileges. [7 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q4 [12]}}