$(a)$ When polythene is rubbed against wool, a number of electrons get transferred from wool to polythene. Hence, wool becomes positively charged and polythene becomes negatively charged.

Amount of charge on the polythene piece, $q=-3 \times 10^{-7} \,C$

Amount of charge on an electron, $e=-1.6 \times 10^{-19}$ $C$

Number of electrons transferred from wool to polythene $=n$

n can be calculated using the relation, $q=n e$

$n=\frac{q}{e}=\frac{-3 \times 10^{-7}}{-1.6 \times 10^{-19}}=1.87 \times 10^{12}$

Therefore, the number of electrons transferred from wool to polythene is $1.87 \times 10^{12}$.

$(b)$ Yes. There is a transfer of mass taking place. This is because an electron has mass, $m _{ e }=9.1 \times 10^{-3}\;kg$

Total mass transferred to polythene from wool, $m = m _{e} \times n$ $=9.1 \times 10^{-31} \times 1.85 \times 10^{12}$

$=1.706 \times 10^{-18}\, kg$

Hence, a negligible amount of mass is transferred from wool to polythene.