consecutive times (or 4,999 times), we expect to wait 500 more attempts until we observe the next success. If, instead, this person focused their attempts on a single safe, and "remembered" their previous attempts to open it, they would be guaranteed to open the safe after, at most, 500 attempts (and, in fact, at onset would only expect to need 250 attempts, not 500). Real-life examples of memorylessness include the universal law of radioactive decay, which describes the time until a given radioactive particle decays, and, potentially, the time until the discovery of a new Bitcoin block, though this has been put in question. An often used (theoretical) example of memorylessness in queueing theory is the time a storekeeper must wait before the arrival of the next customer. Discrete memorylessness Suppose is a discrete random variable whose values lie in the set {0, 1, 2, ...}. The probability distribution of is memoryless precisely if for any and in , we have Here, denotes the conditional probability that the value of is greater than given that it is greater than or equal to . The memoryless discrete probability distributions are the geometric distributions, which count the number of independent, identically distributed Bernoulli trials needed to get one "success". In other words, these are the distributions of waiting time in a Bernoulli process. Note that the above definition applies to the definition of geometric distribution with support {0, 1, 2, ...}. The alternative parameterization with support {1, 2, ...} corresponds to a slightly different definition of discrete memorylessness: namely, that A common misunderstanding "Memorylessness" of the probability distribution of the number of failures before the first success means that, for example, It does mean that which would be true only if the events and were independent, i.e. Continuous memorylessness Suppose is a continuous random variable whose values lie in the non-negative real numbers . The probability distribution of is memoryless precisely if for any non-negative real numbers and , we have This is similar to the discrete version, except that and are constrained only to be non-negative real numbers instead of integers. Rather than counting trials until the first "success", for example, we may be marking time until the arrival of the first phone call at a switchboard. The memoryless distribution is an exponential distribution The only memoryless continuous probability distribution is the exponential distribution, so memorylessness completely characterizes the exponential distribution