Late for my Date!

Alex is having a great time at the Stanford Math Tournament, which is held on February 14th. Then, towards the end, he realizes that he has a date in the evening! (He completely forgot while preparing for the tournament.)

Given the time the tournament ends and the speed at which Alex drives (which is not very fast), he can reach the restaurant where he is supposed to have his date some time between 6:45PM and 7:15 PM.

Unfortunately, he also forgot what time his date should start. All he knows that his date will arrive at the restaurant sometime between 6:30PM and 7:00PM.

Also, Alex knows that his date is impatient and will leave if she has to wait more than 6 minutes for Alex to arrive. As for Alex, he'll wait 12 minutes, and if his date doesn't arrive, he leaves, even if she comes later.

Being an avid math fan, Alex wants to know the probability that he meets with his date. If this probability is represented by \(\frac{m}{n},\) for some relatively prime positive integers \(m\) and \(n\), find \(m + n\).


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