Example 3: HSC Mathematics paper 2002, Q7(c)

Question 7

(c)

Chris has four pairs of socks in a drawer, each pair a different colour.

He selects socks one at a time and at random from the drawer.

(i)

The probability that he does NOT have a matching pair after selecting the second sock is $\frac{6}{7}$. Explain why this is so. 1 mark

(ii)

Find the probability that he does NOT have a matching pair after selecting the third sock. 2 marks

(iii)

What is the probability that the first three socks include a matching pair? 1 mark

ANSWERS

(c)

(i)

After selecting 1 sock, $n(S)=7$. The number of remaining socks that do not match the first sock is 6, ie $n(E)=6$. Probability of selecting a non-matching sock at second selection is

$$P(\mbox{mismatch at }2) = \frac{n(E)}{n(S)} = \frac{6}{7}$$

(ii)

let $M_{i,j}$ be the event that sock from selection $i$ matches the sock from selection $j$ and the event of no match is $\bar M_{i,j}$

$$\begin{eqnarray*} P( \bar M_{1,2} ) & =& \frac{6}{7} \\ P( \bar M_{i,3} \cap \b... ... i=1,2 \\ & = & \frac{4}{6} \times \frac{6}{7} = \frac{4}{7} \end{eqnarray*}$$

(iii)

$$\begin{eqnarray*} P(\mbox{match by selection 3}) & = & 1 - P(\mbox{no match by s... ...\bar M_{1,2}) \right) \\ & = & 1 - \frac{4}{7} = \frac{3}{7} \end{eqnarray*}$$

EXAMINERS' REMARKS, 2002

Question 7

(c)

(i)

Most candidates were unable to explain that there 7 socks left of which 6 did not match the first sock. Elaborate diagrams of coloured socks, tables and tree diagrams were common, as were recalculations of the probability rather than an explanation of how it was obtained. Interestingly, many of the candidates who scored poorly over the whole question were successful here.

(ii)

This part was poorly done with $\frac{6}{7} \times \frac{5}{6}$ as the most common response. Many candidates did not see the link with part (i), while others seemed to be thinking of 3 further selections with answers of the form $\frac{6}{7} \times \frac{5}{6} \times \frac{4}{5}$, which inadvertently led to a correct result.

(iii)

The majority of candidates used the concept of complementary events to obtain their answer, but a large number did not and started the problem all over again.