| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2016 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Direct binomial probability calculation |
| Difficulty | Moderate -0.8 This is a straightforward application of binomial distribution formulas with n=50 and given probabilities. Part (a) requires direct use of binomial probability calculations (some requiring cumulative probabilities), while part (b) is simple recall of mean=np and variance=np(1-p). The only mild challenge is part (iii) requiring recognition that blue or green combines to p=0.5, but overall this is routine S1 material requiring no problem-solving insight. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04d Normal approximation to binomial |
| Colour | Blue | Green | Red | Orange | Yellow | White |
| Proportion | 0.25 | 0.25 | 0.18 | 0.12 | 0.15 | 0.05 |
6 The proportions of different colours of loom bands in a box of 10000 loom bands are given in the table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
Colour & Blue & Green & Red & Orange & Yellow & White \\
\hline
Proportion & 0.25 & 0.25 & 0.18 & 0.12 & 0.15 & 0.05 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item A sample of 50 loom bands is selected at random from the box.
Use a binomial distribution with $n = 50$, together with relevant information from the table, to estimate the probability that this sample contains:
\begin{enumerate}[label=(\roman*)]
\item exactly 4 red loom bands;
\item at most 10 yellow loom bands;
\item at least 30 blue or green loom bands;
\item more than 35 but fewer than 45 loom bands that are neither yellow nor white.
\end{enumerate}\item The random variable $R$ denotes the number of red loom bands in a random sample of $\mathbf { 3 0 0 }$ loom bands selected from the box.
Estimate values for the mean and the variance of $R$.\\[0pt]
[2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA S1 2016 Q6 [12]}}