How to prove P(empty set)=0 from probability axioms step-by-step

📚 Introduction to Probability and the Empty Set

In probability theory, we deal with events and their likelihood of occurring. The probability of an event is a number between 0 and 1, inclusive, where 0 indicates impossibility and 1 indicates certainty. A crucial element in this framework is the concept of the empty set, denoted as $\emptyset$, which represents an event containing no outcomes. Our goal is to rigorously prove that $P(\emptyset) = 0$ using the probability axioms.

📜 Axioms of Probability

The proof relies on the fundamental axioms of probability, established by Andrey Kolmogorov:

• 📏 Axiom 1 (Non-negativity): For any event $A$, $P(A) \geq 0$. This means the probability of any event cannot be negative.
• 💯 Axiom 2 (Normalization): $P(\Omega) = 1$, where $\Omega$ is the sample space (the set of all possible outcomes). This means that the probability of the entire sample space is 1 (certainty).
• ➕ Axiom 3 (Additivity): For any sequence of mutually exclusive (disjoint) events $A_1, A_2, A_3, ...$, $P(A_1 \cup A_2 \cup A_3 \cup ...) = \sum_{i=1}^{\infty} P(A_i)$. This means that the probability of the union of mutually exclusive events is the sum of their individual probabilities.

🔑 Proof that $P(\emptyset) = 0$

1. 🌌 Step 1: Expressing the Sample Space

   We can express the sample space $\Omega$ as the union of itself and the empty set: $\Omega = \Omega \cup \emptyset$. Since $\Omega$ and $\emptyset$ are mutually exclusive (they have no outcomes in common), we can apply Axiom 3.

2. ➕ Step 2: Applying Axiom 3

   Using Axiom 3, we have: $P(\Omega) = P(\Omega \cup \emptyset) = P(\Omega) + P(\emptyset)$.

3. 💯 Step 3: Applying Axiom 2

   From Axiom 2, we know that $P(\Omega) = 1$. Substituting this into the equation from Step 2, we get: $1 = 1 + P(\emptyset)$.

4. ➖ Step 4: Solving for $P(\emptyset)$

   Subtracting 1 from both sides of the equation, we obtain: $0 = P(\emptyset)$.

✅ Conclusion

Therefore, we have rigorously proven, using the axioms of probability, that the probability of the empty set is zero: $P(\emptyset) = 0$.