# 1. Suppose that ? has m clauses and n variables. Prove that the expected number of satisfied…

1. Suppose that ϕ has m clauses and n variables. Prove that the expected number of satisfied clauses under a random truth assignment is at least 7m/8

2. Prove the following general statement about any random variable: Pr [X ≥ E [X]] > 0. (Hint: use conditional expectation.) Then, using this general fact and Exercise 10.159, argue that, for any 3CNF proposition ϕ, there exists a truth assignment that satisfies at least 7 /8 of ϕ’s clauses.

Exercise 10.159

Suppose that ϕ has m clauses and n variables. Prove that the expected number of satisfied clauses under a random truth assignment is at least 7m/8

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