Final answer:
Bonferroni's inequality can be generalized to n events using induction. The inequality states that for n events E₁, E₂, ⋯, Eₙ, the probability of the intersection of all events is greater than or equal to the sum of the individual probabilities minus (n-1).
Step-by-step explanation:
To generalize Bonferroni's inequality to n events using induction, we start by proving the base case for n = 2. Let E₁ and E₂ be two events. We know that P(E₁E₂) = P(E₁) + P(E₂) - P(E₁∪E₂), where ∪ denotes union. The inequality holds true for n = 2.
Next, assume that the inequality holds true for n = k, that is, P(E₁E₂⋯Eₖ) ≥ P(E₁) + P(E₂) + ⋯ + P(Eₖ) - (k-1). Now, we need to prove that it holds true for n = k+1.
Let E₁, E₂, ..., Eₖ be k events. We know that P(E₁E₂⋯EₖEk₊₁) = P(E₁E₂⋯Eₖ) + P(Eₖ₊₁) - P(E₁E₂⋯Eₖ∪Ek₊₁). Using the assumption, we can write:
P(E₁E₂⋯EₖEk₊₁) ≥ [P(E₁) + P(E₂) + ⋯ + P(Eₖ) - (k-1)] + P(Eₖ₊₁) - P(E₁E₂⋯Eₖ∪Ek₊₁).
By rearranging the terms, we get:
P(E₁E₂⋯EₖEk₊₁) ≥ P(E₁) + P(E₂) + ⋯ + P(Eₖ) + P(Eₖ₊₁) - (k-1) - P(E₁E₂⋯Eₖ∪Ek₊₁).
Since (k-1) - P(E₁E₂⋯Eₖ∪Ek₊₁) is a non-negative term, we can rewrite the inequality as:
P(E₁E₂⋯EₖEk₊₁) ≥ P(E₁) + P(E₂) + ⋯ + P(Eₖ) + P(Eₖ₊₁) - (k-1).
Hence, the inequality holds true for n = k+1. By induction, we have proven that the inequality holds for all n events.