smc

The clock will display the incorrect time for the entire hours of $1,10,11$ and $12$. So the correct hour is displayed $\frac{2}{3}$ of the time. The minutes will not display correctly whenever either the tens digit or the ones digit is a $1$, so the minutes that will not display correctly are $10,11,12,\dots ,19$ and $01,21,31,41,$ and $51$. This amounts to fifteen of the sixty possible minutes for any given hour. Hence the fraction of the day that the clock shows the correct time is $\frac{2}{3}\cdot \left(1-\frac{15}{60}\right)=\frac{2}{3}\cdot \frac{3}{4}=\boxed{\frac{1}{2}}$. The answer is $(\mathrm{A})$.