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Let $s_k$ be the $k$-th parking space in a row of $g$ spaces $s_1, s_2, \ldots, s_{g-1}, s_g$. Point $g$ is the entrance of some crowded building, so the closer one is to $g$, the higher the probability of parking spots being occupied (the probability function will monotonically increase, but the formula is unknown). Illustrative function $\frac{1}{1+e^{-(\frac{x}{4}-4)}}$ (Note that the function above, $\frac{1}{1+e^{-(\frac{x}{4}-4)}}$, is illustrative and it is not the only one that satisfies the problem's requirements.)

Suppose a person wants to visit the building. This particular person is quite lazy and tries to minimize walking distance --- that is, parking the car as close to $g$ as possible. S/he advances forward, iterating over $k$, checking in each step if $s_k$ is free or not (so seeing the spots ahead is not an option). When finding a free space, the person must decide whether to park there or move on (advancing to $k+1$).

The optimal situation is to park on the closest-to-$g$ free space. The worst situation would be passing every free space, something undesirable for the driver, who prefers to play it safe: if the probability that $s_k$ is the last free parking space exceeds a certain threshold, s/he will park.

The individual's only source of information for making the forecast is occupation data of observed parking spots, which is recorded.

The person has travelled to spot $s_k$ (because of it, occupation data from $s_1$ to $s_k$ is available) and found it to be free. How could I go about calculating the probability of $s_k$ being the last space available for parking?

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    I'm not sure that your setup supports the problem of "calculating the probability of $s_k$ being the last space available" as well as it would support a problem about the best *strategy* for taking an open space as you pass it. – 2017-02-11
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    @hardmath This might be my statistics ignorance talking, but why is it so? – 2017-02-11
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    As you said, "the probability function will monotonically increase, but the formula is *unknown*". So it seems more likely that one can devise a *strategy* which is optimal with respect to *some objective* given *limited information* about distribution of open parking spaces, than that one can solve for the unknown probability distribution, aka "the probability of $s_k$ being the last space available". – 2017-02-11
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    Guessing is certainly part of statistics and probability (although we prefer the fancy term "estimating"). The issue is not whether we can make a guess, but what properties can be asserted for a particular form of "guess". Formulating the *objective* by which guesses are judged is key to addressing statistical estimation problems. – 2017-02-11
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    There are ways and software to estimate the distribution you gave from discrete points. It can be done from the diagram too but it is much easier if the points are given. – 2017-02-11

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