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A venue acknowledges your order after a delay that is exponentially distributed with mean 10 milliseconds. You have already been waiting 7 milliseconds with no acknowledgment. What is your expected remaining wait in milliseconds?

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The exponential distribution is memoryless. For X sim textExp(lambda), P(X > s + t mid X > s) = fracP(X > s + t)P(X > s) = frace^-lambda(s+t)e^-lambda s = e^-lambda t = P(X > t). Conditional on having waited 7 ms, the remaining wait has exactly the original distribution, so the expected remaining wait is the full mean: E[textremaining] = 10 text ms. The 7 ms already spent is irrelevant. The exponential is the only continuous distribution with this property, which is precisely why it models arrival gaps with no aging: a delay that has survived carries no information about when it ends. The trap is computing 10 - 7 = 3, which silently assumes the delay was a known constant. That logic would be right for a deterministic 10 ms delay and roughly right for a uniform one, but it is exactly wrong here. On a desk this matters for queue and latency models: if fill times are near-exponential, "we have already waited a while, so the fill must be close" is a fallacy, and any system that escalates urgency purely based on elapsed wait is mismodeling the tail.

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