| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Independent binomial samples with compound probability |
| Difficulty | Moderate -0.3 This is a straightforward S2 question testing standard binomial distribution concepts. Part (a) is simple algebra using variance formula np(1-p). Parts (b) use tables directly with minimal manipulation. Parts (c-d) apply basic probability rules (independent events, geometric distribution) but require no novel insight—all are textbook exercises slightly easier than average A-level difficulty. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(B(n, 0.2)\): Var \(= n(0.2)(0.8) = 2.4\) \(n = 15\) | M1 A1 | |
| (b) \(X \sim B(15, 0.2)\) | ||
| (i) \(P(X < 3) = P(X \leq 2) = 0.398\) | B1 M1 A1 | |
| (ii) \(P(X \geq 5) = 1 - P(X \leq 4) = 1 - 0.8358 = 0.164\) | M1 A1 | |
| (c) \(X \sim B(10, 0.8358)\): \(P(X = 10) = 0.8358^{10} = 0.166\) | B1 M1 A1 | |
| (d) \(0.1642^2 \times 0.8358 = 0.00370\) | M1 A1 | 13 marks total |
(a) $B(n, 0.2)$: Var $= n(0.2)(0.8) = 2.4$ $n = 15$ | M1 A1 |
(b) $X \sim B(15, 0.2)$ | |
(i) $P(X < 3) = P(X \leq 2) = 0.398$ | B1 M1 A1 |
(ii) $P(X \geq 5) = 1 - P(X \leq 4) = 1 - 0.8358 = 0.164$ | M1 A1 |
(c) $X \sim B(10, 0.8358)$: $P(X = 10) = 0.8358^{10} = 0.166$ | B1 M1 A1 |
(d) $0.1642^2 \times 0.8358 = 0.00370$ | M1 A1 | 13 marks totalOn a production line, bags are filled with cement and weighed as they emerge. It is found that 20\% of the bags are underweight. In a random sample consisting of $n$ bags, the variance of the number of underweight bags is found to be 2.4.
\begin{enumerate}[label=(\alph*)]
\item Show that $n = 15$. [2 marks]
\item Use cumulative binomial probability tables to find the probability that, in a further random sample of 15 bags, the number that are underweight is
\begin{enumerate}[label=(\roman*)]
\item less than 3, [3 marks]
\item at least 5. [2 marks]
\end{enumerate}
\end{enumerate}
Ten samples of 15 bags each are tested. Find the probability that
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item all these batches contain less than 5 underweight bags, [3 marks]
\item the fourth batch tested is the first to contain less than 5 underweight bags. [3 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q6 [13]}}