By only use these axioms to solves the following two
questions. Thank you.
(AUB)A nB (AnB) AUB 0 EPCA)E P(S)=I PCAUB) P(A) P(B)-PIAne) P(AIB) # ot times A and Boccur #ot times B ocuuts P(ADP(ANB) PCB) P/AB)P(BIA)P(A) P(B) Taew ledr- Using notin The defa P (A I8), The 3 axioms, and T "lews" Teem we have discussed (e. more 1 Show TR P(ALB ) PLACIB) uw-leuti9-2AsSsume AnBUc) (AnB) U (ANC) Mew show Tt Pl(AU B)UC) PIAT+ PCB) Pc) PCANB) PIANC)-...
Prove equalities involving sets A, B, C and D a) (AIB)U(C1B) = (AUC) IB b) (AUB)-(ANB) = (A-8)U(-A) c) (AxB) OLC xD) - (ANC) x (BND) d) (AXB) (BAA) = (ANB)X(AMB)
Exercise 4. By writing AU BUC as (AUB) UC, show that the Principle of Inclusion-Exclusion for three sets is P(AUBUC) = P(A)+P(B)+P(C)- P(ANB) - P(ANC) - P(BNC)+P(ANBNC) Can you generalize the result to an arbitrary number of events?
6. (10 points) Let A, B, and C be sets. Prove (AuB)C(AnC) u(BnC)
3. Let A, B, C be events in a sample space S. Prove that (a) P(AUB) P(A)P(B), (b) P(AUBUC) P(A)+P(B)+P(C)-P(AnB)-P(Anc)-P(Bnc)+P(AnBnc)
Suppose that P(A)0.5, P(B)0.2, P(C) 0.3, P(AnB) 0.1 and P(AnC) 0.1. Compute the following: (a) (2 points) P(AUB) b) (6 points) P(A UC) (c) (4 points) Are the events A and B independent? What about A and C? (d) (8 points) If the sets B and C are mutually exclusive sets, what is P(A U B U C)?
Please help me prove 2,4, and 5. Thank you
Theorem 17. Let A, B and C be sets. Then the following statements are true: (1) AB CA; (2) B CAUB; (3) A CAUB; (4) AB=BA; (5) AU (AUC) = (AUB) UC; (6) An(BNC) = (ANB) nC; (7) An (BUC) = (ANB) U (ANC); (8) AU (BAC) = (AUB) n(AUC).
If P(A)=0.7 ,P(B') = 0.6 and P(AUB)=0.9, then P(ANB) = ? 2. If P(A)=0-7, P(B)=0.6 and P(AUB) - 0.9, then plAl(AUB)) = ? 3 Afair dice is rolled, probability of getting a number & such that 1 <x<8 is = ?
FIGURE 1 (a) What is AC n B in roster notation? (b) What is (BUC) in roster notation? (c) What is AC n(BUC) in roster notation? FIGURE 1 (a) What is An B in roster notation? (b) What is (BUC) in roster Rotation? (c) What is A n(BUC) in roster notation? (d) What is (An B) n(BNC) in roster notation? (e) What is (ANB) U (ANC) in roster notation?
2.1. Use Venn diagrams to verify that (a) (A UB)UC is the same event as AU(BUC) (b) An(BUC) is the same event as (AnB)U(AnC); (c) A U (Br C) is the same event as (A U B) (A U C)