| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Topic | Probability Definitions |
| Type | Combined event algebra |
| Difficulty | Moderate -0.3 This question tests standard probability rules (complement rule, conditional probability) and basic combinatorics (combinations formula). Part (a) requires straightforward application of P(A∪B) = 1 - P(A'∩B') and P(A∩B) = P(A|B)×P(B), while part (b) is direct application of C(49,6). All techniques are routine A-level content with no problem-solving insight required, making it slightly easier than average. |
| Spec | 2.03a Mutually exclusive and independent events2.03d Calculate conditional probability: from first principles5.01a Permutations and combinations: evaluate probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| (a) (i) \(P(A \cup B) = 1 - P(A' \cap B')\) | M1 | |
| \(= \frac{5}{6}\) | A1 | [2] |
| (ii) State \(P(A \cap B) = P(A | B)P(B) = \frac{1}{12}\) | B1 |
| Use \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\) aef | M1 | |
| \(= \frac{2}{3}\) | A1 | [3] |
| (b) \(\frac{49!}{43!6!} = 13\,983\,816\) | M1 A1 | [2] |
(a) (i) $P(A \cup B) = 1 - P(A' \cap B')$ | M1 |
$= \frac{5}{6}$ | A1 | [2]
(ii) State $P(A \cap B) = P(A|B)P(B) = \frac{1}{12}$ | B1 |
Use $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ aef | M1 |
$= \frac{2}{3}$ | A1 | [3]
(b) $\frac{49!}{43!6!} = 13\,983\,816$ | M1 A1 | [2]\begin{enumerate}[label=(\alph*)]
\item Events $A$ and $B$ are such that $\mathrm{P}(A' \cap B') = \frac{1}{6}$.
\begin{enumerate}[label=(\roman*)]
\item Find $\mathrm{P}(A \cup B)$. [2]
\item Given that $\mathrm{P}(A | B) = \frac{1}{4}$ and $\mathrm{P}(B) = \frac{1}{3}$, find $\mathrm{P}(A \cap B)$ and $\mathrm{P}(A)$. [3]
\end{enumerate}
\item In playing the UK Lottery, a set of 6 different integers is chosen irrespective of order from the integers 1 to 49 inclusive. How many different sets of 6 integers can be chosen? [2]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2010 Q12 [7]}}