| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Topic | Conditional Probability |
| Type | Sampling without replacement from bags/boxes |
| Difficulty | Moderate -0.3 This is a straightforward conditional probability question using sampling without replacement. Part (i) and (ii) require basic counting of favorable outcomes over total outcomes, while part (iii) applies the definition of conditional probability P(A|B) = P(A∩B)/P(B) using results from earlier parts. The calculations are routine with no conceptual subtleties, making it slightly easier than average but still requiring careful systematic counting. |
| Spec | 2.03a Mutually exclusive and independent events2.03d Calculate conditional probability: from first principles |
**Question 4**
Answers as fractions need not be fully cancelled down.
**(i)** $P(\text{Same Sex}) = \left(\frac{9}{16} \times \frac{8}{15}\right) + \left(\frac{7}{16} \times \frac{6}{15}\right)$
$= \frac{114}{240} = \frac{19}{40}$ or $0.475$
— M1: One product with correct denominator. M1: Add second product; same denominator. A1: c.a.o. **[3]**
**(ii)** $P(\text{Same sex AND Same year})$
$= \left(\frac{6}{16} \times \frac{5}{15}\right) + \left(\frac{4}{16} \times \frac{3}{15}\right) + \left(\frac{3}{16} \times \frac{2}{15}\right) + \left(\frac{3}{16} \times \frac{2}{15}\right)$
$= \frac{54}{240} = \frac{9}{40}$ or $0.225$
— M1: 4 cases considered; sum of 4 products or terms. A1: All correct. A1: c.a.o. **[3]**
**(iii)** $P(\text{Same year GIVEN Same sex})$
$= \frac{54/240}{114/240} = \frac{9}{19}$ or $0.4736$
— M1: Attempt a quotient of 2 probabilities, with either *their* **(i)** or **(ii)** used correctly. A1: Quotient of $\frac{\textit{their } \textbf{(ii)}}{\textit{their } \textbf{(i)}}$. A1: Ft $\textit{their}\ \frac{\textbf{(ii)}}{\textbf{(i)}}$ provided final answer is between 0 and 1. **[3]**4 At a sixth form college, the student council has 16 members made up as follows. There are 3 male and 3 female students from Year 12, and 6 male and 4 female students from Year 13. Two members of the council are chosen at random to represent the college at conference.
Find the probability that the 2 members chosen are\\
(i) the same sex,\\
(ii) the same sex and from the same year,\\
(iii) from the same year given that they are the same sex.
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2015 Q4 [9]}}