I've been thinking about this for the past few days now, and whatever formula I come up with always fails after a relatively low number. The problem is this:

A button has a 1 in 100 chance of sounding a siren, independent from any other presses. k pigeons are lined in a row, with a button in front of each. All buttons are pressed by the pigeons simultaneously at once per second. When the siren sounds, all pigons will fly away and the buttons will be no longer pressed. After n seconds, what is the chance the pigeons will be still there?

Is there a formula to solve the problem? This is not homework, It has just been bothering me immensely ever since it popped into my mind.