Let Es be an event that occurs with probability 1 – αs for s = 1, 2, …, k. Then prove that...

Let E_s be an event that occurs with probability 1 – α_s for s = 1, 2, …, k. Then prove that

where  is the intersection of the events E₁, E₂, …, E_k. Do not assume that the E_s’s are independent. [This result is called the Bonferroni inequality; see (9.12).] Hint: The proof is by mathematical induction. That is, first show that P(E₁   ∩ E₂) ≥ 1 – α₁ – α₂. Then show that if

is true, the desired result is also true.