Example 7: HSC Mathematics Extension 1 paper, 2001. Q 5

(b)

By using the binomial expansion, show that

$$(q + p)^n - (q-p)^n = 2 \left( \begin{array}{c}n \ 1\end{ar... ... \begin{array}{c}n \ 3\end{array} \right) q^{n-3}p^3 + \ldots$$

What is the last term in the expansion when $n$ is odd?

What is the last term in the expansion when $n$ is even? 3 marks

(c)

A fair sided die is randomly tossed $n$ times.

(i)

Suppose $0 \leq r \leq n$. What is the probability that exactly $r$ ``sixes'' appear in the uppermost position? 2 marks

(ii)

By using the results of part (b), or otherwise, show that the probability that an odd number of ``sixes'' appears is

$${1 \over 2} \left\{ 1 - \left( {2 \over 3} \right)^n \right\} .$$

2 marks

ANSWER

(b)

$$\begin{eqnarray*} (q + p)^n &= & \sum_{k=0}^n \left( \begin{array}{c} n \ k \en... ...begin{array}{c}n \ 3\end{array} \right) q^{n-3}p^3 + \ldots \\ \end{eqnarray*}$$

For $n$ odd, the last term is $2p^n$.
For $n$ even,the last term is $2 nqp^{n-1}$.

(c)

Let $X$ be the number of ``sixes'' and $p$ be the probability of a ``six'' which equals $\frac{1}{6}$. The the probability of $r$ ``sixes'' is

$$P(X=r) = \left( \begin{array}{c} n \ r \end{array} \right) \left[ \frac{1}{6}\right]^r \left[\frac{5}{6}\right]^{n-r}$$

From part (b), we recognise that

$$\mbox{RHS} = 2 \left( P(X=1) + P(X=3) + \ldots \right)$$

Therefore

$$\begin{eqnarray*} P(\mbox{odd number}) & = & \frac{1}{2} \left\{\mbox{LHS of pa... ... & {1 \over 2} \left\{ 1 - \left({ 2 \over 3}\right)^n \right\} \end{eqnarray*}$$

The crux of this problem is defining the problem in terms of the binomial distribution and recognising that the RHS of (b) was double the sum of probabilities of odd numbers of ``successes''.

EXAMINERS' REMARKS, 2001

Question 5

(b)

Most candidates could expand $(q+p)^n$ but to earn more than 1 mark they had to do much more than this. Some candidates seemed confused by the $q$ being first. many simply stated the last terms and this only scored 1 mark. A disappointing number of candidates wrote that the last term is $0$ when $n$ is even.

(c)

(i)

Candidates generally recognised this as a binomial probability and were able to give the correct response. candidates need to clear in their writing, as it was very difficult to distinguish between $n$ and $r$ in many cases.

(ii)

This was very badly done with many candidates not attempting it. Too many candidates could not see the link with (b) or, if they did, they could not clearly show this. Some tried to manipulate the answer to part (i) to get the given expression.