I was given the following exercise:
"At the railway station in London there are three mailboxes: one is labelled as "City", another "Rest of the country" and the last "Abroad". John wants to send three postcards: one to London, another to Cambridge and the last to Rome. He randomly puts a postcard in each mailbox. What is the probability that at least one postcard is correctly mailed?"
I modelled the problem in this way: each letter is represented by a Bernoullian variable, i.e. $$X_i=\begin{cases}1\;\;\;\;\; p=\frac{1}{3}\\ 0\;\;\;\;\; 1-p=\frac{2}{3}\end{cases}.$$ Then, let's call $X=X_1+X_2+X_3$ a Binomial variable. The probability that Jhon correctly sends at least one postcard should be: $$\mathbb P\left(X\ge 1\right)=1-\mathbb P\left(X<1\right)=1-\mathbb P\left(X=0\right)=1-\left[\binom{3}{0}\left(\frac{1}{3}\right)^0\left(\frac{2}{3}\right)^3\right]=\frac{19}{27}.$$ Unfortunately, the right answer is $\frac{2}{3}$. Can you help me finding out where I am making mistake? Thanks in advance.