I have been trying to wrap my head around the memorylessness property of the exponential distribution, and I can't understand the intuition or logic behind it.
Say we have a store where the time between visitors is an exponentially distributed random variable $X$ with $ \lambda = 0.1 $. Observer A starts observing immediately from the time the last visitor left, and according to our distribution, $P(X<20)=0.865, P(X<30)=0.950$.
Say after 20 minutes, there still haven't been any visitors and observer B starts observing the store as well, without prior knowledge of when the last visitor left the store. As we still haven't had any visitors after 20 minutes and the probability of a visitor arriving within 30 minutes is 0.950 observer A expects a high chance of a visitor entering the store within the next 10 minutes. But for observer B the probability of a visitor entering in the next 10 minutes is only $P(X<10)=0.632$. So to me, it seems observers A and B have different probabilities for the same event and just because observer B started observing at a different time point he basically "reset" the probability. This doesn't make any sense, and I assume I must've misunderstood something along the way.
From what I have understood, the memorylessness property tells us in this case that the probability of observers A and B having to wait another 10 minutes is the same, which in my mind invalidates the whole point of probabilities. We calculated the probability of a visitor entering the store within 30 minutes to be 0.950, but if we haven't had any visitors within 20 minutes and the probability of waiting another 10 minutes for both observers is 0.632 then that seems to cancel out the starting probability of waiting 30 minutes for observer A.
Could anybody please help me understand where I have gotten the memoryless property wrong?
Your mistake is the claim that A's probability of observing a visitor in the next 10 minutes equals $\approx 0.950 = \mathbb P(X<30)$. Rather, you should be considering the probability $$\mathbb P(X<30 | X > 20)$$ since it is a known fact from A's perspective that nobody has arrived for the past $20$ minutes. Using Bayes' theorem, you should be able to confirm that $$\mathbb P(X < 30 | X > 20) = \mathbb P(X < 10)\approx 0.632$$ which is precisely the "memorylessness property" of the exponential distribution. It's a special case of the more general memorylessness property $$\mathbb P(X < b | X > a) = \mathbb P(X < b-a)$$