Must... Synchronize... Clocks

Phileas has time-traveled to a new period on Earth where if you ask someone the date, then they will tell you the sum of the month and date, not the two individually. For example, if it was April $13\text{,}$ (which is equal to $04\text{-}13$ ) he would simply be told the number $17$ and nothing more. Luckily for Phileas, he knew exactly what the date was when he was told this information. The probability that this would occur if you traveled to this time period is equal to $\frac{A}{B}\text{,}$ where $A$ and $B$ are positive coprime integers. What is $A+B\text{?}$

$\textbf{Details and Assumptions}$

$-$ Phileas does not know what year it is.

$-$ The new time period still uses the $12\text{-month}$ system that we use today.

$-$ The number of a month is the number that would replace $MM$ if the date was expressed in $MM-DD-YYYY$ form. Similarly, the number of the date is what would replace $DD\text{.}$