We obtain the joint PMF directly from the joint MGF:
[tex]M_{X,Y}(s,t)=0.1+0.2e^s+0.3e^t+0.4e^{s+t}[/tex]
[tex]\implies\mathrm{Pr}[X=x,Y=y]=\begin{cases}0.1&\text{for }x=y=0\\0.2&\text{for }x=1,y=0\\0.3&\text{for }x=0,y=1\\0.4&\text{for }x=y=1\\0&\text{otherwise}\end{cases}[/tex]
Then
[tex]\mathrm{Pr}[X=Y]=\mathrm{Pr}[X=Y=0]+\mathrm{Pr}[X=Y=1]=0.1+0.4=\boxed{0.5}[/tex]
