\( \begin{eqnarray} p_e&=&\frac{2}{3}\frac{p_c}{p_a+p_c}\\ &=&\frac{2}{3}\frac{3p^2(1-p)}{(1-p)^3+3p^2(1-p)}\\ &=&\frac{2p^2(1-p)}{(1-p)^3+3p^2(1-p)}\\ &=&\frac{2p^2\cancel{(1-p)}}{\cancel{(1-p)}\{(1-p)^2+3p^2\}}\\ &=&\frac{2p^2}{(1-p)^2+3p^2} \end{eqnarray} \)

\( \begin{eqnarray} p_e&=&\frac{2p^2}{(1-p)^2+3p^2}\\ &=&\frac{2\times0.01^2}{0.99^2+3\times0.01^2}\\ &=&\frac{2}{99^2+3}\\ &=&\frac{2}{9804}\\ &=&0.00020...\\ &\simeq&0.0002 \end{eqnarray} \)

\( \begin{eqnarray} p_e=\frac{3p^2(1-p)^2+p^4}{(1-p)^4+6p^2(1-p)^2+p^4} \end{eqnarray} \)

\( \begin{eqnarray} p_e=\frac{p^2(3-6p+4p^2)}{1-4p+12p^2-16p^3+8p^4} \end{eqnarray} \)

\( \begin{eqnarray} p_e&=&\frac{3\times0.01^2\times0.99^2+0.01^4}{0.99^4+6\times0.01^2\times0.99^2+0.01^4}\\ &=&\frac{3\times99^2+1}{99^4+6\times99^2+1}\\ &=&\frac{2.94...\times10^4}{9.61...\times10^7}\\ &=&0.00030...\\ &≒&0.0003 \end{eqnarray} \)