$\mathrm{Var}(X+Y) = \mathrm{Var}(X) + \mathrm{Var}(Y)$ ONLY when $X,Y$ are uncorrelated. Otherwise you need Law of Total Variance β interviewers love testing the difference.
Variance of a sum of independent things is the sum of variances. When components depend on each other you carry covariance terms; when they depend on a hidden variable you split with the law of total variance: $\mathrm{Var}(Y) = E[\mathrm{Var}(Y\mid X)] + \mathrm{Var}(E[Y\mid X])$.
β Intro Β· expand
Try first (productive failure)
Before the worked example: spend 60 seconds taking your best shot at this.
A guess is fine β being briefly wrong about a problem makes the explanation
land harder when you read it. This appears once per tutorial; skip
if you already know the trick.
60s
β Try first Β· expand
Worked example
Three independent fair six-sided dice are rolled. What is the variance of their sum?
β Worked example Β· expand
Practice 1 of 3Type a fraction, decimal, or expression β mathjs parses it.
β Practice Β· expand
Reflection
When does the “sum of variances” rule break, and what extra term do you add? Why is the law of total variance often the right move when $Y$ has a hidden parameter?