where N(t) is the number at risk at the beginning of an interval. A hazard is the probability that a patient fails between t {\displaystyle t} and t + Δ t {\displaystyle t+\Delta t} , given that he has survived up to time t {\displaystyle t} , divided by Δ t {\displaystyle \Delta t} , as Δ t {\displaystyle \Delta t} approaches zero.[1]