Let each vehicle arrive at a time uniformly distributed over [0,10). We want the probability that there exists a 2-minute interval [t, t+2) ⊆ [0,10) such that all 6 arrival times fall within [t, t+2).

["Title: Optimal Arrival Distribution for 2-Minute Synchronization: A Probability Insight", "Meta Description:\nExplore the probability that 6 uniformly and independently distributed arrival times in a 10-minute window all fall within some 2-minute interval. Learn the analytical approach and implications for scheduling and logistics.", "---", "### Introduction", "In many operational and logistical settings, seamless coordination—such as synchronized arrivals within narrow time windows—is critical. For example, consider a service hub expecting 6 customer arrivals uniformly distributed over a 10-minute period. A key performance goal is determining the likelihood that all arrivals cluster tightly within some continuous 2-minute interval. Specifically, we want the probability that there exists an interval [t, t+2) ⊆ [0,10) covering all six arrival times.", "This article explores the probability that 6 arrival times, independently and uniformly distributed on [0,10), lie entirely within some 2-minute window. We derive the key mathematical approach, visualize the intuition, and explain practical significance.", "---", "### The Problem: Uniform Arrivals in [0,10) and 2-Minute Coverage", "Let ( T_1, T_2, \dots, T_6 ) be independent random variables uniformly distributed over the interval ([0,10)). A “perfectly synchronized” arrival scenario occurs if there exists a time ( t \in [0,8) ) such that:\n[\nT_i \in [t, t+2) \quad \ ext{for all } i = 1,\dots,6.\n]\nThis means all six arrivals are confined to a 2-minute window, regardless of when that window begins.", "Our goal is to compute:\n[\nP(\exists t \in [0,8) : T_1, \dots, T_6 \subseteq [t, t+2))\n]", "---", "### Intuition: Reformulating the Problem Geometrically", "Since each ( T_i \sim \ ext{Uniform}[0,10) ), we consider the 6D space ([0,10)^6): all possible arrival time combinations. The total volume of this space is ( 10^6 ).", "The event we want corresponds to the volume of the subset where the spread (max − min of arrivals) is less than or equal to 2:\n[\n\max(T_1,\dots,T_6) - \min(T_1,\dots,T_6) \leq 2\n]\nThus:\n[\nP(\ ext{all arrivals fit in some 2-min window}) = \frac{ \ ext{Volume s.t. } \max - \min \leq 2 }{10^6}\n]", "---", "### Mathematical Derivation Using Order Statistics", "Let ( X_{(1)}, X_{(2)}, \dots, X_{(6)} ) denote the order statistics—the sorted arrival times.", "We compute:\n[\nP(\max T_i - \min T_i \leq 2) = P(X_{(6)} - X_{(1)} \leq 2)\n]", "Because the ( T_i ) are i.i.d. uniform on ([0,10)), the joint distribution of order statistics is well known. The joint PDF of ( X_{(1)} ) and ( X_{(6)} ) is:\n[\nf_{X_{(1)},X_{(6)}}(u,s) = 6 \cdot 5 \cdot \left( \frac{s - u}{10} \right)^4 \cdot \mathbf{1}{0 \leq u < s < 10}\n]\nThis arises because:\n- There are ( \binom{6}{2} = 15 ) ways to choose 2 points defining the span,\n- But after ordering, the probability elements distribute over the interval such that the gap constrains how tightly clustered the times are.", "To compute ( P(X \leq 2) ), integrate over the region ( 0 \leq u \leq s \leq u+2 ), ( s \leq 10 ):} - X_{(1)\n[\nP(X_{(6)} - X_{(1)} \leq 2) = \int_{u=0}^{8} \int_{s=u}^{u+2} f_{X_{(1)},X_{(6)}}(u,s),ds,du\n]", "Using the joint distribution:\n[\n= \int_0^8 \int_u^{u+2} 30 \left( \frac{s - u}{10} \right)^4 \cdot \frac{1}{10} ,ds,du\n= \frac{3}{10^5} \int_0^8 \int_u^{u+2} (s - u)^4 ,ds,du\n]", "Evaluate inner integral:\nLet ( v = s - u ), ( dv = ds ), limits from 0 to 2:\n[\n\int_0^2 v^4 ,dv = \frac{32}{5}\n]", "So:\n[\n= \frac{3}{10^5} \int_0^8 \frac{32}{5} ,du = \frac{3}{10^5} \cdot \frac{32}{5} \cdot 8 = \frac{768}{5 \cdot 10^5} = \frac{768}{500000} = 0.001536\n]", "Thus,\n[\nP(\max - \min \leq 2) = 0.001536 = \frac{1536}{1000000} = \frac{24}{15625}\n]", "---", "### Interpretation and Summary", "There is approximately a 0.1536% probability that 6 uniformly distributed arrivals in a 10-minute window will all fit within some continuous 2-minute interval.", "This low probability reflects the challenge of synchronization: while arrivals may look random, the chance they cluster tightly enough to fit in 2 minutes is small due to the large time window.", "---", "### Practical Implications", "This result guides operational decisions:\n- To increase synchronization likelihood, reduce the time window or increase arrival predictability.\n- For logistics, staffing schedules, or shared resource access, guaranteeing tight clustering within short intervals demands repeated arrivals or tighter coordination.\n- Alternatively, consider buffering or extending time windows to match realistic arrival distribution patterns.", "---", "### Conclusion", "Understanding the probability that 6 arrivals fit in some 2-minute interval reveals fundamental constraints in spatial-temporal coordination. By leveraging order statistics and geometric probability, we derived a clean formula reflecting the inherent sparsity of clustering under uniform distributions. This insight helps design better systems where tight arrivals are essential.", "---", "Keywords: uniform distribution, arrival times, synchronization probability, max-min gap, probability in [0,10), logistic modeling, scheduling optimization, order statistics, statistical thresholds."]