9. Suppose (An} are independent events satisfying P(An) < 1, for all n. Show8P(An) = 1 iff P(An i.o. ) = 1.n=1Give an example to show that the condition P(An) < 1 cannot be dropped.

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Answered: 9. Suppose (An) are independent events…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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For part B, explain in detail with proof
5. Let = {x|0 ≤ x ≤ 2}, A = {x | 0.5 < x≤ 1}, B = {x | 0.75 ≤ x < 1.5}, determine the following events: (1) AB; (2) AUB; (3) A B; (4) AB.
2. A number U is selected at random between 0 and 1. Let the events A and B be "U differs from 1/2 by more than 1/4" and "1-U is less than 1/2", respectively. Find the events An B, Ān B, AU B.
i Need Question (d) Only
13. It is given that 5 P (A¬A2) =, P (A2) = - 1 P (A, + A2) = ?, 6' Show that the events Aj and A2 are independent.
1. Events A and B are mutually exclusive with P(A) = 0.280 and P(A or B) = 0.740 a) P(B)= b) P(Not A)= c) P(A and B)=
Assume A and B are independent events with P(A) = 0.42 and P(B) = 0.65. Find (a) P(A n B), (b) P(AUB), (c) P (A'n B'), (d) P (A'n B), and (e) P(ANB'). ... (a) P(An B) = (Type an integer or a decimal.) (b) P(A U B) = (Type an integer or a decimal.) (c) P (A'n B') = (Type an integer or a decimal.) (d) P (A' n B) = (Type an integer or a decimal.) (e) P (An B') = (Type an integer or a decimal.)
Consider 10 independent tosses of a biased coin with the probability of Heads at each toss equal to p, where 0 < p < 1. (a) Let A be the event that there are exactly 6 Heads in the first 8 tosses. Let B be the event that the 9th toss results in Heads. Find P(B | A) and express it in terms of p. (b) Find the probability that there are exactly 3 Heads in the first 4 tosses and exactly 2 Heads in the last 3 tosses. Express your answer in terms of p. (c) Given that there were exactly 4 Heads in the first 7 tosses, find the probability that the 2nd Heads occurred at the 4th toss. Give a numerical answer. (d) Find the probability that there are exactly 5 Heads in the first 8 tosses and exactly 3 Heads in the last 5 tosses.
Let (A,n 2 1) be independent events. Then a)E=1 P(A,) = 0 == P(A i.o.) = 1 b) En=1 P(A,)<=P(A, i.o.) = 0.
Suppose A, B and C are events such that P(A): = 0.46, P(B): = P(BNC) = P(ANC) = 0.16 and P(AnBnC) = 0.06. (a) Are events A and B mutually exclusive? Why? (b) Find P((An BNC)) (c) Find P((AUBUC)) 0.51, P(C) = 0.41, P(An B) =
3 a) The events A and B are such that P(A) =0.3, P(B'| A) =0.8 and P(B| A')=0.4. Find P(AOB) and P(AUB). Hence, state with reason whether A and B are independent events. b) A box contains 5 balls labelled 1, 2, 3, 4 and 5. A player draws a ball from the box randomly. If the number on the ball drawn is 2, 3 or 4, the players score is that number. If the number on the ball is 1 or 5, the player can draw a second ball without replacing the first ball, and his score is the sum of the numbers on the two balls. The events A and B are defined as follows A: The score of a player is 4, 5, 6 or 7. B: A player draws two balls from the box. Find P(A), P(B), P(ANB) and P(AUB).
3. The following is called the Bonferroni's inequality: For events A and B, we have that: P(ANB) ≥ P(A) + P(B) – 1. - a. Prove the Bonferroni inequality. b. Let A and B be events with probabilities P(A) Show that ≤P(AΜB) ≤ } . 12 = 3 and P(B) = 3.

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9. Suppose (An) are independent events satisfying P(An) < 1, for all n. Show 8 P(An) = 1 iff P(An i.o. ) = 1. n=1 Give an example to show that the condition P(An) < 1 cannot be dropped.