Question

Prove using only the axioms of probability that if A and B are events and A...

Prove using only the axioms of probability that if A and B are events and A ⊂ B, then P(Ac ∩ B) = P(B) − P(A).

Homework Answers

Answer #1

Now we know by the additive theorem of probability:

Now A ⊂ B i.e A is a subset of B which means the entire event A is contained in event B which means

Hence

Hence proved.

Let me know in the comments if anything is not clear. I will reply ASAP! Please do upvote if satisfied!

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Q1: a) Re-derive the inclusion-exclusion principle for two events using only the probability axioms. Probability axioms:...
Q1: a) Re-derive the inclusion-exclusion principle for two events using only the probability axioms. Probability axioms: Given an event A in Ω: A1) P(A) >= 0 A2) P(Ω) = 1 A3) P(U (from i=1 to n) A_i) = Σ (from i=1 to n) P(A_i) - if A_i's are disjoint/ mutually exclusive Inclusion Exclusion Principle for two events: (A U B) = (A) + (B) + (A ∩ B) b) Then, using only the axioms and the inclusion-exclusion principle for two...
Prove (a). Events A and B are independent if and only if Ac and Bc are...
Prove (a). Events A and B are independent if and only if Ac and Bc are independent. (b). If events A and B both have a positive probability and are disjoint, then they cannot be independent.
Prove the following using Field Axioms of Real Numbers. prove (b^(−1))^−1=b
Prove the following using Field Axioms of Real Numbers. prove (b^(−1))^−1=b
Let A and B be independent events of some sample space. Using the definition of independence...
Let A and B be independent events of some sample space. Using the definition of independence P(AB) = P(A)P(B), prove that the following events are also independent: (a) A and Bc (b) Ac and B (c) Ac and Bc
Using field and order axioms prove the following theorems: (i) 0 is neither in P nor...
Using field and order axioms prove the following theorems: (i) 0 is neither in P nor in - P (ii) -(-A) = A (where A is a set, as defined in the axioms. (iii) Suppose a and b are elements of R. Then a<=b if and only if a<b or a=b (iv) Let x and y be elements of R. Then either x <= y or y <= x (or both). The order axioms given are : -A = (x...
If A and B are independent events , prove that : 1\ A and B are...
If A and B are independent events , prove that : 1\ A and B are independent 2\ Ac and B are independent   4\Ac and Bc are independent
Use the probability axioms and their consequences on probabilities of events to answer each of the...
Use the probability axioms and their consequences on probabilities of events to answer each of the following. (a) (5 points) Bob believes there is a 50% chance it will rain. He also believes that there is an 80% chance that it will rain and his basement will be flooded. Are these subjective probabilities consistent with the axioms and theorems of probability? Answer YES or NO and explain. (b) (5 points) Suppose 50% of visitors to a museum visit the Chagall...
Using field and order axioms prove the following theorems: (i) Let x, y, and z be...
Using field and order axioms prove the following theorems: (i) Let x, y, and z be elements of R, the a. If 0 < x, and y < z, then xy < xz b. If x < 0 and y < z, then xz < xy (ii) If x, y are elements of R and 0 < x < y, then 0 < y ^ -1 < x ^ -1 (iii) If x,y are elements of R and x <...
For two events A and B show that P (A∩B) ≥ P (A)+P (B)−1. (Hint: Apply...
For two events A and B show that P (A∩B) ≥ P (A)+P (B)−1. (Hint: Apply de Morgan’s law and then the Bonferroni inequality). Derive below Results 1 to 4 from Axioms 1 to 3 given in Section 2.1.2 in the textbook. Result 1: P (Ac) = 1 − P(A) Result 2 : For any two events A and B, P (A∪B) = P (A)+P (B)−P (A∩B) Result 3: For any two events A and B, P(A) = P(A ∩...
Using field axioms and order axioms prove the following theorems (i) The sets R (real numbers),...
Using field axioms and order axioms prove the following theorems (i) The sets R (real numbers), P (positive numbers) and [1, infinity) are all inductive (ii) N (set of natural numbers) is inductive. In particular, 1 is a natural number (iii) If n is a natural number, then n >= 1 (iv) (The induction principle). If M is a subset of N (set of natural numbers) then M = N The following definitions are given: A subset S of R...