I am starting to learn about Poisson processes and I am struggling to put the wording of a question into a probability statement. The questions states that 12 tour boats come per hour and follow a homogenous Poisson process and then asks if you have already been waiting 6 minutes for a tour boat what is the probability you will have to wait another 6 mins for one to arrive.
My initial thought was P(T1 = 12|T1>6) -- T1 being the event time before the first event -- but when going through my notes couldn't find any examples of P(T1= "n") so I then ended up with P(T1<=12|T1>6) however this does not look intuitive to me. If it is correct could you explain why P(T1 <= 12) is correct and if not it would be much appreciated if you could provide some insight on where I'm going wrong.
My understanding is $\mathbb{P}(T\geq 12\mid T \gt 6) = \mathbb{P}(T \gt 6)$ due to the memory-less property.