A particular $24$-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a $2$, it mistakenly displays a $0$. For example, when it is $2:15$ AM the clock incorrectly shows $0:15$ AM, and when it is $22:30$ the clock incorrectly shows $00:30$. What fraction of the day will the clock show the correct time?