A biased coin is weighted such that the probability of obtaining a head is $\frac{4}{7}$. The coin is tossed 6 times and X denotes the number of heads observed. Find the value of the ratio $\frac{{{\text{P}}(X = 3)}}{{{\text{P}}(X = 2)}}$.

recognition of $X \sim {\text{B}}\left( {6,\frac{4}{7}} \right)$     (M1)

${\text{P}}(X = 3) = \left( {\begin{array}{*{20}{c}}   6 \\   3 \end{array}} \right){\left( {\frac{4}{7}} \right)^3}{\left( {\frac{3}{7}} \right)^3}\left( { = 20 \times \frac{{{4^3} \times {3^3}}}{{{7^6}}}} \right)$     A1

${\text{P}}(X = 2) = \left( {\begin{array}{*{20}{c}}   6 \\   3 \end{array}} \right){\left( {\frac{4}{7}} \right)^2}{\left( {\frac{3}{7}} \right)^4}\left( { = 15 \times \frac{{{4^2} \times 34}}{{{7^6}}}} \right)$     A1

$\frac{{{\text{P}}(X = 3)}}{{{\text{P}}(X = 2)}} = \frac{{80}}{{45}}\left( { = \frac{{16}}{9}} \right)$     A1

[4 marks]

Many correct answers were seen to this and the majority of candidates recognised the need to use a Binomial distribution. A number of candidates, although finding the correct expressions for ${\text{P}}(X = 3)$ and ${\text{P}}(X = 4)$, were unable to perform the required simplification.