Are you preparing for a Jane Street quantitative trader interview? One of the classic probability puzzles you may encounter deals with random intervals and their intersection properties. Understanding these questions not only tests your mathematical rigor but also your ability to model and analyze stochastic processes—a skill highly valued at quantitative trading firms like Jane Street. In this article, we’ll break down the famous "Probability of Overlapping Random Intervals" problem, explain key mathematical concepts involved, and guide you step by step through its solution.

Jane Street Quantitative Trader Interview Question: Probability of Overlapping Random Intervals

Problem Statement

Question: Five intervals are independently generated by sampling two points uniformly from [0,1] and taking them as endpoints. What is the probability that all five intervals share a common non-empty intersection?

Key Concepts Involved

• Uniform Distribution
• Random Intervals
• Intersection of Intervals
• Order Statistics
• Probability Calculation

Restating the Problem Mathematically

Let’s formalize the question. For each interval \(I_i\) for \(i = 1, 2, 3, 4, 5\):

• Pick \(X_i, Y_i\) independently and uniformly at random from [0, 1].
• Let \(A_i = \min(X_i, Y_i)\), \(B_i = \max(X_i, Y_i)\), so \(I_i = [A_i, B_i]\).

What is the probability that: \[ \bigcap_{i=1}^5 I_i \neq \emptyset \] That is, all five intervals overlap at some point in [0, 1].

How Are the Random Intervals Generated?

For each interval:

• Pick two numbers \(X\) and \(Y\) independently and uniformly distributed on [0, 1].
• Define the interval as \([A, B]\), where \(A = \min(X, Y)\) and \(B = \max(X, Y)\).
• The random interval is thus a segment of [0, 1], possibly the entire [0, 1] if \(X = 0\) and \(Y = 1\).

Step 1: Reformulation with 10 points

Each of the 5 intervals gives 2 points. So we have 10 i.i.d. uniform points.

We do not label which point is left or right beforehand; the smaller in each pair becomes left, the larger right.

Step 2: Condition for common intersection

For all intervals to intersect, we need:

\(max(left endpoints)≤min(right endpoints)\)

That means there is some point in \([0,1]\) that lies to the right of all left endpoints and to the left of all right endpoints.

Step 3: Key combinatorial observation

If we look at the sorted order of the 10 points, the condition says that among the 10 points, the 5 smallest points must come one from each interval (these become the left endpoints), and the 5 largest points must also come one from each interval (these become the right endpoints).

That is:

• The 5 smallest points are exactly one from each interval.

• The 5 largest points are exactly one from each interval.

Step 4: Counting favorable assignments

Label the intervals 1 to 5. In a random ordering of 10 points coming from 5 pairs:

• Total number of ways to assign the 10 positions in the sorted order to the 5 intervals (each interval appearing twice) is:

 \(10!/(2!)^5\)​

since each interval has 2 identical (in terms of label) points.

Favorable cases
We need the 5 smallest positions to contain exactly one point from each interval. The 5 largest positions will then automatically contain exactly one from each as well, because each interval has exactly 2 points total.

Number of favorable assignments:

• Assign the 5 smallest positions: choose a permutation of the 5 intervals: \(5!\) ways.

• Assign the 5 largest positions: also a permutation of the 5 intervals: \(5!\) ways.

So total favorable = \(5!.5!\)

Step 5: Probability

\(P = 5!⋅5!/ 10!(2!)^5= 8/63\)

Python Simulation

Let’s verify the result with

import numpy as np

def random_interval():
    x, y = np.random.uniform(0, 1, 2)
    return min(x, y), max(x, y)

def all_overlap(n_intervals=5):
    intervals = [random_interval() for _ in range(n_intervals)]
    left = max(interval[0] for interval in intervals)
    right = min(interval[1] for interval in intervals)
    return left <= right

# Run the simulation
trials = 1_000_000
successes = sum(all_overlap(5) for _ in range(trials))
probability = successes / trials

print(f"Estimated probability: {probability:.6f}")

Jane Street Interview Tips

Quantitative interviews at Jane Street and similar trading firms often feature probability and combinatorial puzzles like this. Here are some tips for tackling these:

• Clarify and formalize the problem. Restate the question in precise mathematical terms.
• Draw diagrams. Visualizing intervals and their overlap can help you find the right approach.
• Look for symmetry and invariance. Many intervals problems are greatly simplified by symmetry arguments.
• Identify relevant distributions. In this problem, understanding uniform and order statistics is essential.
• Generalize. Try to solve for \( n \) intervals, not just the specific value given. This shows depth of understanding.

Conclusion

The "Probability of Overlapping Random Intervals" problem is a beautiful illustration of how combinatorial geometry and probability theory intersect. For five random intervals generated by selecting two independent uniform points on [0, 1] and using them as endpoints, the probability that all intervals overlap is:

\[ \boxed{\frac{8}{63}} \]

Problems like these are common in Jane Street quantitative interviews because they test your ability to model random processes, think abstractly, and communicate mathematical arguments clearly. Practice with similar questions, verify your answers computationally, and approach every step methodically. Good luck with your interviews!

References

• Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1.
• Jane Street Sample Interview Questions: Official Jane Street Careers
• Probability Stack Exchange: Probability of overlap of n random intervals
• Wikipedia: Beta function

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