| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2021 |
| Session | June |
| Marks | 11 |
| Topic | Permutations & Arrangements |
| Type | Specific items separated |
| Difficulty | Challenging +1.2 This is a multi-part combinatorics problem requiring systematic case analysis and careful counting. Part (a) involves straightforward hypergeometric probability with 'at least' requiring summing cases. Part (b)(i) uses the standard 'treat as a block' technique with verification. Part (b)(ii) requires more sophisticated reasoning to avoid overcounting when splitting vowels into two groups. While methodical, it demands careful organization and is above average difficulty for A-level, though these techniques are standard in Further Statistics. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
26 cards are each labelled with a different letter of the alphabet, A to Z. The letters A, E, I, O and U are vowels.
\begin{enumerate}[label=(\alph*)]
\item Five cards are selected at random without replacement.
Determine the probability that the letters on at least three of the cards are vowels. [4]
\item All 26 cards are arranged in a line, in random order.
\begin{enumerate}[label=(\roman*)]
\item Show that the probability that all the vowels are next to one another is $\frac{1}{2990}$. [3]
\item Determine the probability that three of the vowels are next to each other, and the other two vowels are next to each other, but the five vowels are not all next to each other. [4]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics 2021 Q3 [11]}}