| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | At least one success |
| Difficulty | Standard +0.3 This is a straightforward binomial distribution question requiring identification of the model, basic probability calculations, and a simple inequality involving logarithms. The ratio calculation (p=1/10 for red, p=4/10 for blue) is routine, and all parts follow standard S2 techniques with no novel problem-solving required. Part (d) involves a standard 'at least one' complement approach with logarithms, which is a common S2 exercise slightly above pure recall. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| (a) binomial, \(n = 10\), \(p = \frac{1}{2}\) | B2 | |
| (b) \(p\) would vary | B1 | |
| (c) (i) let \(X\) = no. of blue beads \(\therefore X \sim B(10, \frac{1}{2})\); \(P(X = 5) = 0.6230 - 0.3770 = 0.2460\) [0.2461 (4sf) using \({}^{10}C_5...\)] | M1, A1 | |
| (ii) let \(Y\) = no. of red beads \(\therefore Y \sim B(10, \frac{1}{2})\); \(P(X > 0) = 1 - P(X = 0) = 1 - (\frac{1}{2})^{10} = 0.7369\) (4sf) | M1, M1, M1, A1 | |
| (d) let \(R\) = no. of red beads in \(n\) picks \(\therefore R \sim B(n, \frac{1}{2})\); \(P(R > 0) > 0.99\) \(\therefore P(R = 0) < 0.01\) \(\therefore (\frac{1}{2})^n < \frac{1}{100}\) | M2, A1 | (12 marks) |
**(a)** binomial, $n = 10$, $p = \frac{1}{2}$ | B2 |
**(b)** $p$ would vary | B1 |
**(c)** (i) let $X$ = no. of blue beads $\therefore X \sim B(10, \frac{1}{2})$; $P(X = 5) = 0.6230 - 0.3770 = 0.2460$ [0.2461 (4sf) using ${}^{10}C_5...$] | M1, A1 |
(ii) let $Y$ = no. of red beads $\therefore Y \sim B(10, \frac{1}{2})$; $P(X > 0) = 1 - P(X = 0) = 1 - (\frac{1}{2})^{10} = 0.7369$ (4sf) | M1, M1, M1, A1 |
**(d)** let $R$ = no. of red beads in $n$ picks $\therefore R \sim B(n, \frac{1}{2})$; $P(R > 0) > 0.99$ $\therefore P(R = 0) < 0.01$ $\therefore (\frac{1}{2})^n < \frac{1}{100}$ | M2, A1 | (12 marks)A bag contains 40 beads of the same shape and size. The ratio of red to green to blue beads is $1 : 3 : 4$ and there are no beads of any other colour.
In an experiment, a bead is picked at random, its colour noted and the bead replaced in the bag. This is done ten times.
\begin{enumerate}[label=(\alph*)]
\item Suggest a suitable distribution for modelling the number of times a blue bead is picked out and give the value of any parameters needed. [2]
\item Explain why this distribution would not be suitable if the beads were not replaced in the bag. [1]
\item Find the probability that of the ten beads picked out
\begin{enumerate}[label=(\roman*)]
\item five are blue,
\item at least one is red. [6]
\end{enumerate}
\end{enumerate}
The experiment is repeated, but this time a bead is picked out and replaced $n$ times.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Find in the form $a^n < b$, where $a$ and $b$ are exact fractions, the condition which $n$ must satisfy in order to have at least a 99\% chance of picking out at least one red bead. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 Q4 [12]}}