(a) Let (A, :n E N) be a sequence of events in some sample space 2 Set A- (wefl:wEA, infinitely often) and B- (wel:we A, for all sufficiently large n). (i) Show that B-U A (ii) Show that A is an event and that P(A) < P(A). for all integers n. () Suppose that the series P(A.) converges. Show that P(A)0.

Q: (a) Prove that for events A and B, if P(B|A) > P(B) then P(A|B) > P(A).

A: As per our guidelines, we are allowed to answer first question only. Thanks   Given Events A and B…

Q: 4 Prove Borel Cantelli theorem (lecture notes p.16) i.e. Let (2, F, P) be a probability space and…

A: Convergent definition in mathematics is a property (displayed by certain innumerable series and…

Q: Prove that a sequence of random variables converges almost surely

A: Definition: A sequence of random variables X1​,X2​,X3​,… converges almost surely (a.s.) to a random…

Q: 2. Consider a system of five identic al components connected in series, as illustrated in the figure…

A: System can fail when any one of the components fails. So total possible outcomes of system fails…

Q: Suppose that events M and N are mutually exclusive, P(M)= 0.6, and P(N) = 0.8 What is P(M and N)?…

A: P(M)= 0.6, and P(N) = 0.8

Q: Let A and B be events with P(A) = 3/10, P(B) = 4/10), and P(AUB) = 6/10. (a) Compute P(AC). (b)…

A: From the provided information, P(A) = 3/10, P(B) = 4/10, and P(A U B) = 6/10

Q: a) The expression tree for an algebraic expression is given below. Apply an inorder tree traversal…

A: We have to apply in order tree transversal to find the value of the algebraic expression. Here we…

Q: A system consists of five identical components connected in series as shown: 2 3 -5 As soon as one…

A:

Q: Suppose E and F are events such that n(E) = 14, n(F) = 15, and n(ENF) = 2. a) Find P(E|F). b) Find…

A: Givenn(E)=14n(F)=15n(E∩F)=2Total no. of events(n)=14+15-2=27

Q: A card is drawn at random from a standard 52-card deck. Events G and H are G= the drawn card is…

A: Total card = 52 are G= the drawn card is black H= the drawn card is divisible by 4 (face cards are…

Q: What is the critical value to test the hypothesis at the 5% level of significance that the training…

A: It is given that  Sample size, n = 15 Significance level, α = 0.05

Q: 3. A fair coin is tossed four times, and the sequence of heads and tails is observed. a) List each…

A: A = {at least 3 heads}B = {at most 2 heads}C = {heads on the third toss}D = {1 head and 3 tails}

Q: (4) Consider a roll of six dice of different colors. (a) How many different rolls are possible?…

A: Six dice of different colors are rolled.

Q: 3. If {An, n ≥ 1) is an independent sequence of events, show P(An) = P(An). μ| n=1 n=1

A: Given: Let  An,n≥1 is an independent sequence of events. We need to show that; P∩n=1∞An=∏n=1∞PAn

Q: a) Two events, M and N, are such that P(M) = 0.6, P(M NN) = 0.2, P(M UN) = 0.85 Find P(N \M) State,…

A:

Q: Suppose that for events A, B, we have: P(A) = 1/6 , P(B) = 5/12 , P(A|B) + P(B|A) = 7/10. What is P(…

A:

Q: 6 involving independent and disjoint events. For this problem, assume that Pr[A∪B]=0.55 and…

A:

Q: Let A and B be events. Show that each of the following is true. (a) 1+ P(An B) ≥ P(A) + P(B) (b) P…

A: We have to prove the given statements for the probability of events A and B.

Q: |20. Suppose a fair coin is tossed three times. Let A be the event that a head appears on the first…

A:

Q: Suppose that A and B are events such that P(A) = 1/3, P(B) = 1/5, and P(A|B) + P(B|A) = 2/3. What is…

A: It is given that P(A)=1/3, P(B)=1/5 and P(A|B)+P(B|A)=2/3.

Q: The Maclaurin series for f(x)=3¬** is Select one: O In3 (1+ x? + 2! x6. x8 . .) 3! 4! 0 1+In3 x +…

A: The Maclaurin's Series is given by fx=f0+xf'0+x22!f''0+x33!f'''0+.....

Q: Find the sum of the series S. 3- 3 In(8) 3 (In(8))? - 3 (In(8))³ _ ... 2! 3!

A:

Q: If A and B are events, with P(A) = (Simplify your answer.) 15 and P(B) = 2 3 1 and P(A and B) = then…

A: Solution-: Given: P(A) =15, P(B)=23 and P(A and B) =13 We wan to find P(A or B)=?

Q: Find the series shown. 2(3 n- 4) n=2

A: We can find the answer as below

Q: (a) Set gn = f − fn and translate the preceding hypothesis into statementsabout the sequence (gn).

A:

Q: P(AB)=P(BC)=0, P(AC)=1/8. TH O A.1/2 O B.5/8

A: The formula of P(AUBUC) is shown as follows: P(AUBUC)=P(A)+P(B)+P(C)-P(AB)-P(BC)-P(AC)+P(A∩B∩C) We…

Q: Q1

A:

Q: Let (Ω, F, P) be a probability space. If E1, Ε2,..., En €F are events, show that for all positive…

A: Given that We have to show that

Q: (C) Assuming that B represents boy and G represents girl, which sequence of children is more…

A: Given B represents Boy and G represents a girl. the given sequences are 1) BGBGB2)BGGGG The…

Q: Property 3. (Mean of the composite series ). If xi (i = 1, 2,..., k ) are the means of k component…

A:

Q: Q1. With 4 tosses of a coin the possible sequence of outcomes is listed below: (1=Heads, 0= Tails):…

A: A fair coin contains two outcomes i.e., Heads &amp; Tails. Like, if a fair coin tossed n times,…

Q: (a) In a class of 20 students, one student is chosen president, another one is chosen secretary,…

A: NOTE: As per the guidelines, we are supposed to solve the first question only. We see that,  A…

Q: Find the sum of the series S. 3 - 3 In(3) + 3 (In(3))² - 3 (In(3))² ... 2! 3! S =|| !!

A: Sum of series

Q: A system with n components is operating in series. Suppose each component has the same probability…

A:

Q: A fair coin is tossed three times. Let R, S and T be three events such that R:maximum one coin shows…

A: Sample space={HHH,HHT,HTH,THH,TTT,TTH,THT,TTH} R={HHT,HTH,THH,HHH} S={HHH}…

Q: aclaurin series for f(x)%3= one: 6' +13 13! i6 x2 4. x2 9 + - 5. 13 +13 5! 7!

A: Given function is, f(x)=sinh(x2)x

Q: Number 6 only

A: Event:  The collection of outcomes in an experiment is called as an event. Independent Events: Any…

Q: Given  P(A) = 0.3, P(b) = 0.5 and P( A and B ) = 0.2.       Are the events A,B independent ?…

A: Thank you for posting the question. Since you have asked multiple question(Different questions), we…

Q: Statement. For n events A, Az, ..., Ag, we have A₂, 'n' n (α) 0) P(24) 2 | > ΣΡ(4)-(n-1) i=1 η ομώνα…

A:

Q: 3. (a) A city has sunny weather on 3 out of every 5 days on average. Use the binomial formula to…

A: (a) Given Data: The probability of having a sunny day is: p=3/5 The expression to calculate the…

Q: Consider two events A and B such that P(A) = 1/2 and P(B) = 1/3. Find P (A n BC) under each of the…

A: P(A)=1/2, P(B)=1/3

Q: Establish what is meant by (a) a pair of mutually exclusive events and (b) a pair of statistically…

A: Solution: 6. Let A be the event that a smoker will suffer from chronic respiratory illness and B be…