One day, at precisely 9:30 AM, Willy loses Arman's Student ID card. It turns out that the ID was lost in Dr. Wu's Algebra class. Dr. Wu, confused, puts the ID on the board and draws an arrow saying "Arman's ID".

Willy knows he has made a tremendous mistake, so he immediately tells Arman. Enraged, Arman gives Willy a warning and then searches the school for his ID.

It is given that Arman will go to Dr. Wu's classroom at a random time between 10 AM and 11 AM and see his ID. However, Dr. Wu will also check his board at a random time between 10 AM and 11 AM. If he sees that the ID is not there, he will not worry about it, but if he sees the ID still there, he will wait 15 more minutes for Arman to come and then throw it away.

The probability that Arman's ID is thrown away can be expressed as a common fraction \(\frac{a}{b}\). Find the product \(ab\).

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