Consider a recurring event for which the time periods between consecutive events is exponentially distributed. For argument's sake, I'm waiting for a taxi on a busy street. How might one calculate the likelihood of a taxi showing up in the next second?

Intuitively the chance that the event will occur in the next second should be just the integral of the PDF over the range (NOW, NOW+1). However, the fact that I am waiting for the event during this time period is not independent of the fact that the event did not occur in the time period (0, NOW]. Thus, perhaps I should consider the the integral of the PDF over the range [0, NOW+1) to get the probability of the event to occur in the next second. Though this "feels wrong" it does make sense as for each given second, the likelihood of the event occurring in that second is higher than it will be for the following second, given that the event has not already occurred.