We have 3 independent Bernoulli random variables with parameter 1/2. They are pairwise independent but not mutually independent. I know that the variance of Var[A] = Var[B] = Var [C] = 0.25 due to the formula of Bernoulli random variables variance: p * (1-p). Therefore, I know that Var[A] + Var[B] + Var [C] = 0.75. I know as well that Var[A + B + C] = 0.75 because Var[A + B + C] = Var[A] + Var[B] + Var [C] when the variables are pairwise independent.
However, I am not able to calculate by myself Var[A + B + C]. Can you please guide mr algebraically?
Thanks
$$\mathrm{Var}(A+B+C)=\mathrm{Cov}(A+B+C,A+B+C)=\mathrm{Var}(A)+\mathrm{Var}(B)+\mathrm{Var}(C)+2\mathrm{Cov}(A,B)+2\mathrm{Cov}(A,C)+2\mathrm{Cov}(B,C)$$ In order for the variance of the sum to be the sum of the variance you don't even need anything as strong as pairwise independence; it is enough for the covariance of each pair to be $0$.