Expectation with exponential distribution

Question

Suppose I arrive at a cafe with 3 baristas. All 3 baristas are busy but no one is in the queue. So I’ll wait until one of the baristas finish serving their current customer and then give my order to the barista that gets freed the first. Suppose, the service time of every order for all 3 baristas is a random variable with exponential distribution with mean 3 minutes. Let X be the random variable that denotes my total waiting time from the moment I enter the cafe until I receive my order. Note that the service time for any order is independent. i. (5 points) Find E[X].

The correct answer is E[X] = 1 + 3 = 4

My confusion is where did the 1 come from?

Answer 1

as soon as a barista is freed up to serve you, you wait 3 minutes

the 1 is expected time for a barista to be freed up, for each of the 3 it's about 1.5 minutes

but 1 minute for one, 1.5 for a 2nd, and 2 for a 3rd, average = (1+1.5+2)/3= 1.5

you get the 1st with 1 minute. add 3 to get E(x)= 1+3= 4 minutes