\color{rgb(0,0,255)}P\left( X=k \right)=\left( \begin{array}{l} 4 \\1 \end{array} \right)0.25 ^1\times \left( 1-0.25 \right)^3 = 4\times 0.25 ^1\times 0.75 ^3\approx 0.421875
$$\color{rgb(127,0,0)}S_{1}$$
$$\color{rgb(127,0,0)}\overline {S_{1}}$$
$$\color{rgb(127,0,0)}S_{2}$$
$$\color{rgb(127,0,0)}\overline {S_{2}}$$
$$\color{rgb(127,0,0)}S_{3}$$
$$\color{rgb(127,0,0)}\overline {S_{3}}$$
$$\color{rgb(127,0,0)}S_{4}$$
$$\color{rgb(127,0,0)}\overline {S_{4}}$$
$$\color{rgb(127,0,0)}S_{4}$$
$$\color{rgb(127,0,0)}\overline {S_{4}}$$
$$\color{rgb(127,0,0)}S_{3}$$
$$\color{rgb(127,0,0)}\overline {S_{3}}$$
$$\color{rgb(127,0,0)}S_{4}$$
$$\color{rgb(127,0,0)}\overline {S_{4}}$$
$$\color{rgb(127,0,0)}S_{4}$$
$$\color{rgb(127,0,0)}\overline {S_{4}}$$
$$\color{rgb(127,0,0)}S_{2}$$
$$\color{rgb(127,0,0)}\overline {S_{2}}$$
$$\color{rgb(127,0,0)}S_{3}$$
$$\color{rgb(127,0,0)}\overline {S_{3}}$$
$$\color{rgb(127,0,0)}S_{4}$$
$$\color{rgb(127,0,0)}\overline {S_{4}}$$
$$\color{rgb(127,0,0)}S_{4}$$
$$\color{rgb(127,0,0)}\overline {S_{4}}$$
$$\color{rgb(127,0,0)}S_{3}$$
$$\color{rgb(127,0,0)}\overline {S_{3}}$$
$$\color{rgb(127,0,0)}S_{4}$$
$$\color{rgb(127,0,0)}\overline {S_{4}}$$
$$\color{rgb(127,0,0)}S_{4}$$
$$\color{rgb(127,0,0)}\overline {S_{4}}$$
$$\color{rgb(0,0,0)}\textcolor{black}{\rightarrow 0.25 ^{4}\times 0.75 ^{0}=0.00390625 }$$
$$\color{rgb(0,0,0)}\textcolor{black}{\rightarrow 0.25 ^{3}\times 0.75 ^{1}=0.01171875 }$$
$$\color{rgb(0,0,0)}\textcolor{black}{\rightarrow 0.25 ^{3}\times 0.75 ^{1}=0.01171875 }$$
$$\color{rgb(0,0,0)}\textcolor{black}{\rightarrow 0.25 ^{2}\times 0.75 ^{2}=0.03515625 }$$
$$\color{rgb(0,0,0)}\textcolor{black}{\rightarrow 0.25 ^{3}\times 0.75 ^{1}=0.01171875 }$$
$$\color{rgb(0,0,0)}\textcolor{black}{\rightarrow 0.25 ^{2}\times 0.75 ^{2}=0.03515625 }$$
$$\color{rgb(0,0,0)}\textcolor{black}{\rightarrow 0.25 ^{2}\times 0.75 ^{2}=0.03515625 }$$
$$\color{rgb(0,0,0)}\textcolor{red}{\rightarrow 0.25 ^{1}\times 0.75 ^{3}=0.10546875 }$$
$$\color{rgb(0,0,0)}\textcolor{black}{\rightarrow 0.25 ^{3}\times 0.75 ^{1}=0.01171875 }$$
$$\color{rgb(0,0,0)}\textcolor{black}{\rightarrow 0.25 ^{2}\times 0.75 ^{2}=0.03515625 }$$
$$\color{rgb(0,0,0)}\textcolor{black}{\rightarrow 0.25 ^{2}\times 0.75 ^{2}=0.03515625 }$$
$$\color{rgb(0,0,0)}\textcolor{red}{\rightarrow 0.25 ^{1}\times 0.75 ^{3}=0.10546875 }$$
$$\color{rgb(0,0,0)}\textcolor{black}{\rightarrow 0.25 ^{2}\times 0.75 ^{2}=0.03515625 }$$
$$\color{rgb(0,0,0)}\textcolor{red}{\rightarrow 0.25 ^{1}\times 0.75 ^{3}=0.10546875 }$$
$$\color{rgb(0,0,0)}\textcolor{red}{\rightarrow 0.25 ^{1}\times 0.75 ^{3}=0.10546875 }$$
$$\color{rgb(0,0,0)}\textcolor{black}{\rightarrow 0.25 ^{0}\times 0.75 ^{4}=0.31640625 }$$