Question

(a) State the three axioms of probability, and using just these axioms, prove that for any events A, B, PAB) = P(B) - P(ANB). (b) State the definition of the concept of the independence of three events A, B and C. Prove that if A, B and C are independent, then A and B nē are independent. (c) What are the necessary properties of any event space? (d) "The equality P(AUB) P(A)P(B) + P(B) implies that A and B are independent." Is this statement true or false? Justify your answer.

Answer 1

The three axioms of probability are non-negativity, additivity, and normalization. Using these axioms, we can prove that P(A n B) = P(B) - P(A U B).

Step-by-step explanation:

The three axioms of probability are:

1. Non-negativity: For any event A, the probability of A is greater than or equal to zero: P(A) >= 0.
2. Additivity: For any two mutually exclusive events A and B, the probability of the union of A and B is equal to the sum of their individual probabilities: P(A U B) = P(A) + P(B).
3. Normalization: The probability of the entire sample space S is equal to one: P(S) = 1.

To prove that P(A n B) = P(B) - P(A U B), we can use the following steps:

1. Start with the equation P(A U B) = P(A) + P(B) - P(A n B) (from the additivity axiom).
2. Rearrange the equation to isolate P(A n B): P(A n B) = P(A) + P(B) - P(A U B).
3. Substitute P(A U B) with 1 (from the normalization axiom): P(A n B) = P(A) + P(B) - 1.
4. Simplify the equation: P(A n B) = P(B) - P(A U B).

Therefore, we have proved that P(A n B) = P(B) - P(A U B) using just the three axioms of probability.