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Compute Parking Clustering Probability and P-Value

GustoData Scientist · Technical Screen
MediumData ScientistTechnical ScreenStatistics & Math

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This question tests your ability to calculate exact combinatorial probabilities under a null hypothesis and define a p-value for a spatial clustering test.

The Parking Spot Problem

Imagine a linear parking lot with N total spaces arranged in a single row. There are K cars parked in these spaces, where each car is assigned to exactly one space, and no two cars occupy the same space. Assume that all possible arrangements of the K cars are equally likely (the null hypothesis of spatial randomness).

A "cluster" is defined as a sequence of 2 or more adjacent occupied spaces. For example, if N=5 and K=3, the arrangement [1, 1, 1, 0, 0] contains one cluster of size 3, while [1, 0, 1, 0, 1] contains zero clusters.

Tasks

  1. Given N spaces and K cars, what is the total number of ways to arrange the cars?
  2. Suppose we observe M specific clusters in a given configuration. How would you define the p-value for this observation under the null hypothesis?
  3. For N=5 and K=2, calculate the exact probability of observing at least one cluster of size 2.

Examples

  • N=3, K=2: Possible arrangements: [1,1,0], [1,0,1], [0,1,1]. Arrangements with at least one cluster: [1,1,0] and [0,1,1]. Probability: 2/3.

Constraints

  • 1 <= K < N
  • Assume K and N are small integers for exact calculation.

Hints

Hint 1

The total number of ways to choose K positions out of N is given by the binomial coefficient 'N choose K'.

Hint 2

Under the null hypothesis, the p-value is the probability of observing a test statistic as extreme or more extreme than the one observed, given the distribution of all possible arrangements.

Hint 3

For part 3, list all possible subsets of size 2 from the 5 available spots and count how many have the indices i and i+1 present.

Follow-up questions

  • How does the probability change if the parking lot is circular (i.e., space 1 is adjacent to space N)?
  • If N and K are very large, how would you approximate this probability distribution using a normal distribution?
  • What are the implications for your p-value if the occupancy is not uniform but follows a Poisson process?