Q10. Prove that if A, B and C are independent, then AUB (A union B) is independentAdd Image/PDF files to Q10of C. Q9. If A and B are two events, prove that P(ANB) ≥ 1 − P(Ā) – P(B). [Note: Thisis a simplified version of the Bonferroni inequality.]-

In probability theory, two events are said to be independent if the occurrence of one does not…

Answered: Q10. Prove that if A, B and C are…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.8: Probability

Problem 32E

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Assume that the probability that an airplane engine will fail during a torture test is 12and that the aircraft in question has 4 engines. Construct a sample space for the torture test. Use S for survive and F for fail.
Q9. If A and B are two events, prove that P(ANB) ≥ 1 − P(Ā) – P(B). [Note: This is a simplified version of the Bonferroni inequality.]
For any three events A, B, & D. Such that Pr(D)>0. Prove that Pr(A U B|D) = Pr(A|D) ÷ Pr(B|D) - Pr(A n B|D)
There are 2 players in a game. Each player independently picks a real number on the interval [0, 1] uniformly. If the (absolute) difference between the two numbers is less than a (0 < a < 1) and the sum of the two numbers is less than 1, then both players win, otherwise both players lose. What is the probability of winning? 2.
(b) Suppose the 4-digit PINS are randomly distributed. How many peo- ple must be in a room such that the probability that two of them have the same PIN is at least 1/2? (Here "4 digits" means that the PIN cannot start with a 0.) Note: Pay special attention to the direction of the inequality in the formula(s) used for this question.
6. The prosecutor's fallacy. Let G be the event that some accused person is guilty, and I the event that some testimony or evidence presented is in fact true. It has been known for lawyers to argue on the assumption that P(G|T) = P(TIG). Show that this holds if and only if P(G)=P(7).
Q16
A container contains 45 green tokens, 5 blue tokens, and 3 red tokens. Two tokens are randomly selected without replacement. Compute P(F|E). E−you select a blue token first F−the second token is blue
Q1. A chest of drawers has 3 drawers. Each drawer has 2 boxes. The boxes of one drawer contain a silver coin in each respectively, the boxes of another a gold coin in each box, and the boxes of the third drawer a gold and a silver coin, respectively. A drawer is selected at random and a box from the drawer is selected at random and opened. The coin is found to be silver. What is the probability that the coin in the other box is gold? (Harder Problem)
Let X - Geom(p). Let s 2 0 be an integer. Show that P(X > s) = (1 – p)s. (Hint: The probability that more than s trials are needed to obtain the first success is equal to the probability that the first s trials are all failures.) a. b. Let t 2 0 be an integer. Show that P(X >s + 1 | X > s) = P(X > t). This is the memoryless property .[Hint: P(X > s + t and X > s) = P(X > s + t).] A penny and a nickel are both fair coins. The penny is tossed three times and comes up tails each time. Now both coins will be tossed twice each, so that the penny will be tossed a total of five times and the nickel will be tossed twice. Use the memoryless property to compute the conditional probability that all five tosses of the penny will be tails, given that the first three tosses were tails. Then compute the probability that both tosses of the nickel will be tails. Are both probabilities the same? C.
A container contains 50 green tokens, 15 blue tokens, and 4 red tokens. Two tokens are randomly selected without replacement. Compute P(F|E). E−you select a non−green token first F−the second token is gereen P(F|E)=
Could you double-check if my answers are correct for this question 2.17? If they are not correct, please point out the error and present the correction. Question. 2.17: a. P(A) + P(B) does not necessarily equal 1 because the events A and B could potentially overlap. In other words, it's possible for a request to be related to both the SPSS package and SAS simultaneously. The formula P(A) + P(B) - P(A ∩ B) gives the correct probability of either event A or event B occurring, taking into account the possibility of overlap. b. The complement of event A, denoted as A', is the event that the next request is not related to the SPSS package. The probability of the complement of A can be calculated as: P(A') = 1 - P(A) = 1 - 0.30 = 0.70 c. The union of events A and B, denoted as A ∪ B, is the event that the next request is related to either the SPSS package or SAS or both. The probability of the union of A and B can be calculated using the formula: P(A ∪ B) = P(A)…

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Q10. Prove that if A, B and C are independent, then AUB (A union B) is independent Add Image/PDF files to Q10 of C. Q9. If A and B are two events, prove that P(ANB) ≥ 1 − P(Ā) – P(B). [Note: This is a simplified version of the Bonferroni inequality.] -