Restaurant is sending orders every 5 minutes on average.
Question: From 15:00 until 15:05, two orders were ordered. What's the probability that the next order will be ordered until 15:10 ?
I can conclude that's it's about Exponential distribution with $\lambda = 12$, because our time unit is hours, and we have 12 orders in one hour.
I'm not sure how to refer to "From 15:00 until 15:05, two orders were ordered".
But without that info. I can tell that the probability that the next order will be ordered is less than 5 minutes, thus $Pr[X \le 1/12]$.
Recall that the exponential distribution has the property of "memorylessness". This means previous orders do not increase or decrease the probability of new orders.
What is the probability that someone orders in the next five minutes?
That is indeed Exp(12,1/12) (Exponential distribution with $\lambda = 12$ and time 1/12 = 5 minutes). Hope this helps