Proving Boole's Inequality Without Using Induction

The Boole's Inequality (also known as the Jensen's Inequality for Countable Partitions) is a fundamental statement in probability theory. It asserts that for any finite collection of events in a probability space, the probability of their union is bounded above by the sum of their individual probabilities. Specifically, for events (A_1, A_2, ldots, A_n), the inequality is:

[Pleft(bigcup_{i1}^n A_iright) leq sum_{i1}^n PA_i]

Step-by-Step Proof of Boole's Inequality

To prove Boole's Inequality without using induction, we can rely on the principle of inclusion-exclusion and the properties of probabilities. Here’s a detailed step-by-step proof:

Step 1: Define the Events and Their Probabilities

Let (A_1, A_2, ldots, A_n) be events in a probability space. Our goal is to show that the probability of the union of these events is less than or equal to the sum of the probabilities of the individual events.

Step 2: Use Inclusion-Exclusion

The probability of the union of the events can be expressed using the principle of inclusion-exclusion:

[Pleft(bigcup_{i1}^n A_iright) sum_{i1}^n P(A_i) - sum_{1 leq i

Step 3: Analyze the Terms

Notice that the terms in the inclusion-exclusion formula alternate in sign. Each (P(A_i cap A_j)) and higher-order intersections are subtracted from the sum of the probabilities of individual events. It is crucial to recognize that all intersection probabilities (P(A_i cap A_j)) and higher-order intersections are non-negative because probabilities cannot be negative. Therefore, when we subtract these intersection terms from the sum of the individual probabilities, we are not adding anything that would increase the total.

Mathematically, this can be represented as:

[Pleft(bigcup_{i1}^n A_iright) sum_{i1}^n P(A_i) - sum_{1 leq i

Since all intersection terms are non-negative, the right-hand side will be less than or equal to the sum of the probabilities of the individual events. Thus:

[Pleft(bigcup_{i1}^n A_iright) leq sum_{i1}^n P(A_i)]

Step 4: Conclude the Proof

Therefore, we have shown that the probability of the union of the events is less than or equal to the sum of the probabilities of the individual events, which is precisely Boole's Inequality.

Summary

In summary, Boole's Inequality can be proven by applying the inclusion-exclusion principle and recognizing that all the terms subtracted from the sum of individual probabilities are non-negative. This leads directly to the conclusion required by the inequality.

Key Takeaways:

Boole's Inequality states that the probability of the union of events is less than or equal to the sum of their probabilities. The principle of inclusion-exclusion is crucial for the proof. Probabilities of intersections cannot be negative, hence they do not increase the total when subtracted.

By understanding and applying these principles, one can effectively navigate the intricacies of probability theory and use Boole's Inequality in various real-world scenarios.