Memoryless Waiting Times, explained.
The memoryless property makes a remaining waiting-time distribution independent of elapsed waiting under the relevant conditioning.
Why it happens
Independent attempts succeed with fixed probability 0.2. Conditional failure for the next k turns remains 0.8 to power k after any declared number of preceding failures.
The memoryless property makes a remaining waiting-time distribution independent of elapsed waiting under the relevant conditioning.
Read the result
The chance of initially reaching a long wait becomes small, while the conditional next-attempt probability stays fixed under independence.
A worked example
Separate elapsed rarity from remaining odds
Set already observed failures to zero and inspect the next-attempt probability twenty percent.
Raise previous failures to thirty while keeping the next-three-attempt window fixed.
Read the unchanged chance 48.8 percent of at least one success in those three future attempts.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
Independent attempts succeed with fixed probability 0.2. Conditional failure for the next k turns remains 0.8 to power k after any declared number of preceding failures.
Where this idea is useful
A practical use
Separate how unlikely a long initial wait was from the probability of success on the next independent attempt.
A common misconception
“A long unsuccessful wait makes an independent success due.”
The conditional future distribution stays the same for geometric attempts; the rarity of the observed past does not change the next independent trial.
What this explanation leaves out
- The model is geometric with fixed independent attempts. Ageing equipment, queues and scheduled events need not be memoryless.
Does memorylessness describe worn-out equipment?
Not automatically. Age-dependent failure rates violate the constant-probability model, making elapsed age relevant information.
Which assumption makes elapsed waiting irrelevant to the remaining distribution?
Associated thinkers
Associations marked provisional are awaiting source review.
Further reading
Explore the original research or the teaching reference behind this experiment.