50 people each purchase a ticket for concert A and a ticket for concert B. The concerts both occur in an auditorium of 50 seats. If the seat numbers for the tickets are distributed randomly, what's the probability that no one has the same seat number for both concerts?
I tried using random variables: Each person has a $(\frac{1}{50})(\frac{49}{50})$ chance of getting differing seat numbers, so the probability of no one getting the same seat number for both concerts is $(50)(\frac{49}{2500}) = \frac{49}{50}$. However, I am concerned about how the seat numbers are dependent; person X occupying seat 1 in concert A means that no one else can take that seat... is this a valid concern?