In a relay race with 4 teams each consisting of 4 runners, if the runners are randomly assigned to run the 1st, 2nd, 3rd, or 4th leg of the race, what is the probability that at least one team has its runners running in ascending order of their ages?
Instruction: Each of four teams has four runners with distinct ages. Assign each team a uniformly random leg order, independently across teams. Calculate the probability that at least one team runs in strictly increasing age order.
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Official answer
Assume each team's four runners have distinct ages, all 24 orders within a team are equally likely, and the four team orderings are independent. Exactly one order per team is strictly increasing by age, so its probability is 1/24.
The chance no team has increasing order is (23/24)^4. Therefore
P(at least one) = 1−(23/24)^4 = 51935/331776 ≈ 0.156536, or 15.65%.
The formula does not evaluate to 14.2%. If ages can tie, define whether “ascending” permits equal ages and recount favorable orders. If assignments are coupled across teams, the complement need not factor into four identical probabilities.