I Never Knew There Are 21 Hours In A Day

I look at my 21-hour digital watch twice a day, once in the morning (at a random time between $00:01$ and $10:30$ , inclusive) and once in the afternoon (at a random time from $10:31$ to $21:00$ , inclusive). If the probability that at least once on this day I see the time $a:b$ , where $b$ is a multiple or factor of $a$ , can be expressed as $\frac{A}{B}$ , where $A$ and $B$ are positive coprime integers, find the value of $A+B$ .

Details and Assumptions:

• My watch shows only hours and minutes. There are 60 minutes in an hour.
• 0 is not a factor or multiple of any number.
• $b$ can be both a factor and multiple of $a$ .
• We are in the Ordovician period of the Earth's history. The second of six periods in the Paleozoic era. There are 21 hours in a day, and a year lasts 417 days, because the Earth spins faster on its axis. Not that it matters.

This is original.

P.S.: During the Ordovician there is less oxygen in the air than nowadays, so one would have to use an oxygen mask, as Nigel Marven does here: