the memoryless property of the exponential distribution

You have already been waiting twenty minutes for a bus that is exponentially distributed. Does your long wait so far mean the bus is 'due' any moment, or that you have used up your bad luck? Surprisingly, no. The exponential distribution forgets the past completely: the remaining wait looks exactly like a fresh wait, as if you had just arrived. That is its memoryless property.

Formally, P(X > s + t given X > s) = P(X > t) for all s, t >= 0. Read it as: given you have already survived past time s, the chance you survive a further t is the same as the original chance of surviving t from the start. The proof is one line: P(X > s + t and X > s)/P(X > s) = e^(-lambda(s+t))/e^(-lambda s) = e^(-lambda t), because the e^(-lambda s) factors cancel. The exponential is the ONLY continuous distribution with this property (the geometric is the only discrete one). Its constant hazard rate is the same fact in disguise: the instantaneous risk of the event never changes with age.

Memorylessness is exactly why the exponential models things that do not wear out — a phone call queue, a radioactive nucleus, a 'random' failure with no aging mechanism. It is also a powerful caveat: it is the wrong model whenever waiting longer changes the odds. A part that fatigues, a person whose risk rises with age, a coupon that nears expiry — none of these are memoryless. Mistaking the gambler's-fallacy intuition ('I'm overdue') for the math is the classic error; the exponential is the math that says you are never overdue.