| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Independent binomial samples with compound probability |
| Difficulty | Standard +0.3 This is a standard S2 binomial distribution question with straightforward application of formulas. Parts (a)-(b) use basic binomial probability with n=10, p=0.05. Parts (c)-(d) involve a compound binomial (binomial of binomials) but with clear structure. Part (e) requires normal approximation to binomial, a routine S2 technique. All parts follow textbook patterns with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04d Normal approximation to binomial |
| Answer | Marks |
|---|---|
| (a) No. of blemished apples \(\sim B(10, 0.05)\); from tables, \(P(X \geq 2) = 1 - 0.9139 = 0.0861\) | B1; M1 A1 |
| (b) \(P(X = 2) = 0.9885 - 0.9139 = 0.0746\) | M1 M1 A1 |
| (c) Now \(X \sim B(50, 0.0861)\); \(P(\text{no trays}) = 0.9139^{50} = 0.0111\) | B1 M1; M1 A1 |
| \(P(\text{1 tray}) = 50 \times 0.9139^{49} \times 0.0861 = 0.0522\) | M1 A1 |
| So \(P(X < 2) = 0.0111 + 0.0522 = 0.0633\) | A1 |
| (d) \(50 \times 0.0861 = 4.3\), so expect 4 trays | M1 A1 |
| (e) No. blemished in 20 trays \(\sim B(200, 0.05) = \text{Po}(10)\) | B1 |
| \(P(X > 10) = 1 - 0.4579 = 0.542\) | M1 A1 |
(a) No. of blemished apples $\sim B(10, 0.05)$; from tables, $P(X \geq 2) = 1 - 0.9139 = 0.0861$ | B1; M1 A1 |
(b) $P(X = 2) = 0.9885 - 0.9139 = 0.0746$ | M1 M1 A1 |
(c) Now $X \sim B(50, 0.0861)$; $P(\text{no trays}) = 0.9139^{50} = 0.0111$ | B1 M1; M1 A1 |
$P(\text{1 tray}) = 50 \times 0.9139^{49} \times 0.0861 = 0.0522$ | M1 A1 |
So $P(X < 2) = 0.0111 + 0.0522 = 0.0633$ | A1 |
(d) $50 \times 0.0861 = 4.3$, so expect 4 trays | M1 A1 |
(e) No. blemished in 20 trays $\sim B(200, 0.05) = \text{Po}(10)$ | B1 |
$P(X > 10) = 1 - 0.4579 = 0.542$ | M1 A1 |
**Total: 16 marks**
---6. In a fruit packing plant, apples are packed on to trays of 10 , and then checked for blemishes. The chance of any particular apple having a blemish is $5 \%$. If a tray is selected at random, find
\begin{enumerate}[label=(\alph*)]
\item the probability that at least two of the apples in it are blemished,
\item the probability that exactly two are blemished.
Trays are now packed in boxes of 50 trays each. In one such box, find
\item the probability that at most one tray contains at least two blemished apples,
\item the expected number of trays containing at least two blemished apples.
\item Use a suitable approximation to find the probability that in a random selection of 20 trays there are more than 10 blemished apples.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q6 [16]}}