Suppose we choose a random number $b$ from the interval $[0,1]$ with a uniform distribution. Based on this number, we define the following random variables:$$X_n=\begin{cases}1 & b\in \left(\frac{1}{2n},\frac{1}{2n}+\frac{1}{3}\right)\\0 & \text{otherwise}\end{cases}$$
These random variables are Bernoulli random variables with an expected value of $\frac{1}{3}$. Are they independent? Explain your answer.
I think they are not independent. For example, if we know that $X_1=1$, then we can be sure that $X_4=0$. So they are dependent on each other. But, when calculating $\operatorname{var}(X_1+X_2+\dots+X_n)$, I noticed that my instructor used $n\operatorname{var}(X_1)$. How did he get this result, using covariance or something else? How?
I appreciate any help!