extbook 2.23: If A and B are events and B C A, why is it “obvious" that P(B) < P(A)? [hint: Use the ct that we proved in the class A = (AN B)U(AnB) to prove that A = BU(ANB) ifBCA and Axioms

Q: 1. Let B1, ... , Bn be independent events. Show P(UB:) = 1 – [[(1 – P(B;)). %3D i=1 i=1

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Q: 1. An experiment consists of asking 3 women at random if they wash their dishes with brand X…

A: A closed-ended question has been asked to three respondents and their responses are recorded in the…

Q: Consider four independent events A1, A2, A3, and A4, andlet pi 5 P(Ai) for i 5 1,2,3,4. Express the…

A: Let pi be the probability of the event Ai. Denote qi=1- pi as the probability that event Ai does not…

Q: 1. Assume that X is a discrete random variable with E(X) = -2, E(X²)= 9, E(X) = -5. and %3D %3D…

A: E(X)=-2E(X2)=9E(X3)=-5

Q: . P(E complement)

A: Here it is given that, P(E) = 0.60 P(F) = 0.15

Q: 1. Let A and B be events such that, P(A) = 0.6, P(B) = 0.7, P(A N B) = 0.4 Find %3D a. P(A U B) b.…

A: Given that, A and B are events, PA=0.6PB=0.7PA∩B=0.4

Q: 2. Let A and B be two events with P(A) = and P(B) = . Show that %3D (a) P(AU B) > 3/4 (b) 를<P(An B)…

A: Event: In probability theory, an event is an outcome or a set of outcomes of a random experiment to…

Q: Prove that P(A') = 1 − P(A), for any event A

A:

Q: 4. Show that an event A is independent of itself iff P(A) is either 0 or 1.

A: Proof: PA=PA∩A=PAPA  Since A is independent itself.⇒PA1-PA=0PA=1  Since P(A)&gt;0

Q: If E and F are mutually exclusive events with P(E) = 0.11 and P(F) 0.48, then the value of P(EUF…

A:

Q: 5. Let A,. A. A, be three independent events. Prove or disprove that Af. Aş, and A are independent.

A:

Q: Suppose that a random variable X :S → R is such that the set X(S) is countably infinite, that is,…

A: We want to find the sum of P(X=x) is equal to

Q: For two standard dice all 36 outcomes of a throw are equally likely. Find P(X1 + X2 = j) for all j…

A:

Q: Let E and F be two events in S with P(E) = 0.7, P(F) = 0.26, and P(E U F) = 0.8. Find P(En F) and…

A:

Q: If a random variable X is defined such that: E[(X - 3)2] = 18 and E[(X-1)²] = 34. Then the values of…

A:

Q: Q4. Prove that for any three events A, B and C (Inclusion-Exclusion Principle). P(A U B) = P(A) +…

A: If A and B are two events (subsets of a sample space S) and are not disjoint, then we have to show…

Q: Q2. Prove that P(A') = 1 – P(A), for any event A.

A: Hi! Thank you for the question, As per the honor code, we are allowed to answer one question at a…

Q: Let E and F be two events in S with P(E) = 0.39, P(F) = 0.4, and P(E ∩ F) = 0.22. Find P(EC ∩ F) and…

A: To compute the probabilities of the required events , using addition and complementation laws

Q: THEOREM 3.3. If X EXP (^,), Y` EXP 2), X and Yare in- dependent, and A + A2, then Z = X + Y has a…

A:

Q: Page 2 of 9 1. Suppose A, B, and C are independent events in some proability space, and that P(A) =…

A: It is given that P(A) =0.3, P(B)=0.4 and P(C)=0.5.

Q: Let E and F be two events that are mutually exclusive and suppose P( E) = .4 and P(F) = .2. Compute…

A: Solution: Mutually exclusive events: Two events said to be disjoint when the happening of two events…

Q: 5. A certain drugstore has three public telephone booths. For i = 0, 1, 2, 3, let p; denote the…

A: A certain drugstore has three public telephone booths. Given that the probability that ith public…

Q: Let B1, ... , Bn be independent events. Show n n P(ÜB.) = 1- T1c (1— Р(B,)). i=1 1=1

A: P∪I=1nBi=1-∏i=0n1-PBi For n =1             R.H.S=1-(1-PB1)=1-1+PB1=PB1=L.H.S For n = 2…

Q: Q2. Prove that P(A^') = 1 − P(A), for any event A

A: It is an important part of statistics . It is widely used .

Q: Show that if E1,... , En are events such that P(E;) = 1 for i = 1, ...,n, then Ek) = 1. k=1

A: We have events E1, E2,...., En such that PEi = 1. We have to that the probability of the event…

Q: Siven that P(A) = 0.06, P(B) = 0.23, and P(A and B) = 0.0258, are events A and B ndependent? %3D %3D…

A:

Q: 10. Let A and B be events such that Pr[B] = 0.7,Pr[A U B] = 0.9, Pr[A|B] = 0.3. Find Pr [A]. %3D %3D

A: Provided information: P(B)=0.7P(A∪B)=0.9P(A|B)=0.3

Q: Are these events mutually exclusive (E) or mutually inclusive (1)? Blank 1 1. Rolling an even and an…

A:

Q: (b) Let E and E, be events with IP(E,) > 0 and P(E2) > 0. Prove that if P(E E2)> P(E1) then P(E2|E1)…

A: Event: In probability theory, an event is an outcome or a set of outcomes of a random experiment to…

Q: 1. Let A and B be two events such that P(A) = 0.4, P(B) = 0.7, P(A U B) = 0.9 %3D a) Find P(A n B).

A: Given: P (A) = 0.4 P (B) = 0.7 P (A U B) = 0.9   a) P (A ∩ B) = ?

Q: Let E and F be two events in Swith P(E) = 0.3, P(F) = 0.5, and P(EN F) = 0.2. Find P(EN F). (Hint:…

A:

Q: Q3.The relay network in figüre, operates if and only if there is a closed path of relays from left…

A: Given: The network operates only if there is a closed path of relays from left to right. Consider…

Q: If A and B are independent events such that P(A )=0.7 and P(B)=0.5 then P(A U B) O a. 0.9 O b. 0.5…

A: Probability: If a random experiment results in n mutually exclusive and equally likely outcomes; out…

Q: Given that P(A) = 0.47, P(B) = 0.07, and P(A and B) = 0.0329, are events A and B %3D %3D %3D…

A: Solution:- From given information p(A)=0.47p(B)=0.07p(A and B)=0.0329

Q: If A is any event from sample sapace S then P(A)+P(A)=

A: Here we need to find the sum of an event and it's compliment.

Q: Pe 14. (i) A and B are two events such that P(A) = 1/4, P(B) = 1/3, and P(AUB) = 1/2 ind P(BLA). %3D…

A: Given data P(A) = 1/4 P(B) = 1/3 PA∪B = 1/2 PB|A = PA and B/P(A) P(A U B) = P(A) + P(B) - P(A and B)…

Q: Theorem 3-7. For n events A, Az, ..., Ag, we have P(U a) = E P(A.) – EE P(A, n A) + EE P(A;nA;nAd)…

A:

Q: Let E, and E, be events with IP(E1) > 0 and P(E2) > 0. Prove that if P(E, E2) > P(E1) then P(E2|E1)…

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Q: Theorem 2.1.5 If A, A2 ..., A, are n ( z 2) events, then P UA i- 1 E P(A;} – E P(A¡ N A}} + £ P(A¡N…

A:

Q: Let n e N, c > 0. Suppose that A1,..., An and B1,...,Bn are two families of independent events in a…

A: Given :A1,...,An and B1,...,Bn are two families of independent events in aprobability space Bk⊂Ak.To…

Q: 5. Consider three events: E1, E2, and F. Assume that P(F) > 0, and that P(E, N E2|F) =…

A: Introduction:  Conditional Probability: Let there are two events , event A and event B. Then the…

Q: 26. If A1,..., A, are events, define S = P(A;) i=1 S2 = P(A;Aj) 1si<jsn S3 = E P(A;A;Ar) Σ 1si<j<ksn…

A:

Q: Theorem 3-8. For n events A1, A2, ..., A„ 11 P( 4)> U A; 2 E P (A;) – E P (A;^ A) i = 1 i = 1…

A:

Q: 5. Let A and B be events such that 0 < P(A) < 1 and 0 < P(B) < 1 a. Show that if A and B are…

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Q: 6) Show that, if X and Y are independent random variables, such that E|X| < co ,and B is an…

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Q: et F1, F2, and F3 are pairwise disjoint events from a sample space = F;UF2UF3. Let E be an event in…

A:

Q: 1. Let A and B be independent events with P (A) = 0.7 and P (B) = 0.2. Compute: (a) P (A∩B) (b) P…

A: Given: PA=0.7PB=0.2 A and B are independent events. 1. (a) P (A∩B) is obtained as below:…

Q: Let E1,.., En be independent events. Prove that P(E1 U..U En) = 1 – 1[[1 – P(E:)]. i=1

A: Definitions used: 1. Complementation: P(AC)=1-P(A) 2. P(A∪B)C=P(AC∩BC) 3. If A and B are independent…

Q: Prove that if E1, E2,.., En are independent events, then P(E, U E2 U .. U En) = 1- | 1– P(E;)] i=1

A: Given that, E1, E2, ........., En are independent events. We need to prove that: P(E1∪E2 ∪…