已知$X\sim B\left(n,p\right)$，若$4P\left(X=2\right)=3P\left(X=3\right)$，则$p$的最大值为$ $ $(\qquad)$ $ $.

$\\dfrac{5}{6}$

$\\dfrac{4}{5}$

$\\dfrac{3}{4}$

$\\dfrac{2}{3}$

$\because 4P\left(X=2\right)=3P\left(X=3\right)$，
$\therefore 4{\rm C}_n^2{p^2}{(1-p)^{n-2}}=3{\rm C}_n^3{p^3}{(1-p)^{n-3}}$，整理得$4\left(1-p\right)=\left(n-2\right)p$，即$p=\dfrac{4}{{n+2}}$，
又$\because n\in {\bf N}^{*}$，且$n\geqslant 3$，
$\therefore p\leqslant\dfrac{4}{5}$，
$\therefore p$的最大值为$\dfrac{4}{5}$.
故选：$\rm B $