| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2003 |
| Session | January |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Independent binomial samples with compound probability |
| Difficulty | Standard +0.3 This is a straightforward S2 question testing standard binomial distribution calculations (parts a-c), normal approximation to binomial (part d), and basic normal distribution (part e). All parts use direct formula application with no novel problem-solving required, though part (c) requires recognizing a compound binomial structure (binomial of binomials), making it slightly above average difficulty for typical A-level. |
| Spec | 2.04d Normal approximation to binomial2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance5.02d Binomial: mean np and variance np(1-p) |
| Answer | Marks | Guidance |
|---|---|---|
| Let \(X\) represent the number of double yolks in a box of eggs; \(X \sim B(12, 0.05)\) | B1 B1 | |
| \(P(X = 1) = P(X \leq 1) - P(X \leq 0) = 0.8816 - 0.5404 = 0.3412\) | M1 A1 | (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X > 3) = 1 - P(X \leq 3) = 1 - 0.9978 = 0.0022\) | M1 A1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(\text{only } 2) = C_2^3(0.3412)^2(0.6588)^2\) | M1 A1 | |
| \(= 0.230087\) | A1 | (3 marks) |
| Answer | Marks |
|---|---|
| Let \(X\) represent number of double yolks in 10 dozen eggs; \(X \sim B(120, 0.05) \Rightarrow X \approx \text{Po}(6)\) | B1 |
| \(P(X \geq 9) = 1 - P(X \leq 8) = 1 - 0.8472\) | M1 A1 |
| \(= 0.1528\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Let \(X\) represent the weight of an egg; \(W \sim N(65,\ 2.4^2)\) | M1 | |
| \(P(X > 68) = P\!\left(Z > \frac{68 - 65}{2.4}\right)\) | A1 | |
| \(= P(Z > 1.25)\) | A1 | |
| \(= 0.1056\) | A1 | (3 marks) |
# Question 5:
## Part (a)
| Let $X$ represent the number of double yolks in a box of eggs; $X \sim B(12, 0.05)$ | B1 B1 | |
| $P(X = 1) = P(X \leq 1) - P(X \leq 0) = 0.8816 - 0.5404 = 0.3412$ | M1 A1 | (3 marks) |
## Part (b)
| $P(X > 3) = 1 - P(X \leq 3) = 1 - 0.9978 = 0.0022$ | M1 A1 | (2 marks) |
## Part (c)
| $P(\text{only } 2) = C_2^3(0.3412)^2(0.6588)^2$ | M1 A1 | |
| $= 0.230087$ | A1 | (3 marks) |
## Part (d)
| Let $X$ represent number of double yolks in 10 dozen eggs; $X \sim B(120, 0.05) \Rightarrow X \approx \text{Po}(6)$ | B1 | |
| $P(X \geq 9) = 1 - P(X \leq 8) = 1 - 0.8472$ | M1 A1 | |
| $= 0.1528$ | A1 | |
## Part (e)
| Let $X$ represent the weight of an egg; $W \sim N(65,\ 2.4^2)$ | M1 | |
| $P(X > 68) = P\!\left(Z > \frac{68 - 65}{2.4}\right)$ | A1 | |
| $= P(Z > 1.25)$ | A1 | |
| $= 0.1056$ | A1 | (3 marks) |
---5. A farmer noticed that some of the eggs laid by his hens had double yolks. He estimated the probability of this happening to be 0.05 . Eggs are packed in boxes of 12 .
Find the probability that in a box, the number of eggs with double yolks will be
\begin{enumerate}[label=(\alph*)]
\item exactly one,
\item more than three.
A customer bought three boxes.
\item Find the probability that only 2 of the boxes contained exactly 1 egg with a double yolk.
The farmer delivered 10 boxes to a local shop.
\item Using a suitable approximation, find the probability that the delivery contained at least 9 eggs with double yolks.
The weight of an individual egg can be modelled by a normal distribution with mean 65 g and standard deviation 2.4 g .
\item Find the probability that a randomly chosen egg weighs more than 68 g .
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2003 Q5 [15]}}