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Can someone please explain how this double sum works and why the LHS = RHS?

$$2 \sum_{1\leq j < k \leq N} P(F_j F_k) = \sum_{j\neq k} P(F_j F_k)$$

user477465
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    The right-hand side includes all terms with $j\ne k$, so it includes things like $P(F_1 F_3)$ and also $P(F_3 F_1)$. These terms are equal, so we can just keep one of them (i.e. just sum over all indices where the first index is strictly less than the second), and then double the result. The left-hand side does this. – Minus One-Twelfth Mar 02 '19 at 06:02
  • thank you, i finally understand – user477465 Mar 02 '19 at 06:06

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We have $$F_j \cap F_k = F_k \cap F_j,$$

Hence

$$P(F_j\cap F_k)+P(F_k \cap F_j)=2P(F_j\cap F_k)$$

Siong Thye Goh
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