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📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Solution: We are given 5 distinct transitions occurring at unique seconds in a 3600-second interval, with all orderings of transition times uniformly random. We are to compute the probability that two specific transitions, say $T_1$ and $T_2$, occur at consecutive seconds in the sequence — that is, one immediately follows the other (either $T_1$ then $T_2$, or $T_2$ then $T_1$).

📌 First, the total number of ways to assign 5 distinct transition times to 3600 seconds such that they are at distinct seconds is equivalent to choosing 5 distinct time points and permuting them in order:

📌 But since we are computing a probability over uniformly random orderings of the 5 transitions (with fixed distinct timing positions), we can instead consider the $5!$ possible orderings of the transitions, with each ordering equally likely under the uniform model.

📌 Now, focus on the two specific transitions. Among all $5!$ orderings, we count how many have these two consecutive in the sequence, regardless of position.

📌 We treat the pair of specific transitions as a block. There are $2$ ways to order this block: $A$ then $B$, or $B$ then $A$. The block and the remaining 3 transitions form 4 entities to permute:

📌 Total number of orderings: $5! = 120$

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