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Prove that if 0 < p < 1 the following is a probability
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- *2.100 Show that Theorem 2.6, the additive law of probability, holds for conditional probabilities. That is, if A, B, and C are events such that P(C) > 0, prove that P(A U B|C) = P(A|C) + P(B|C)–P(ANB|C). [Hint: Make use of the distributive law (AUB)NC = (ANC)U(BNC).] The Additive Law of Probability The probability of the union of two events A and B is THEOREM 2.6 P(AUB) = P(A) + P(B) – P(AN B). If A and B are mutually exclusive events, P(AN B) = 0 and P(AU B) = P(A)+ P(B).Let A and B be events such that Pr[A] =0.370.37, Pr[B] =0.630.63, and Pr[A ∩ B] =0.240.24. Find Pr[A|B].Find the value of z for the following probability statement P(-1.9Maui played a game of darts where a player gets to throw six darts to burst balloons. Basedon the performance of the previous customers, there is a 40% chance that a dart will hit aballoon. Let X be the number of balloons successfully hit by Maui. Define a function that willgenerate the probability of X. Show that it is indeed a probability function using the following axioms:Denote by a* 0 < x <1 0.5 2 <*Number 21113 -The random variable X takes the values 0, 1 and 2 with probabilities 1/4, 2/4 and 1/4, respectively. Find the value of the cumulative probability function F (X <2 and X = 2) of the discrete probability division in question. A)0 B)1/4 C)one D)3/4 TO)1/2The central limit theorem says that: lima to0 P[ < æ] = ® (x)Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons IncProbability and Statistics for Engineering and th…StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage LearningStatistics for The Behavioral Sciences (MindTap C…StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage LearningElementary Statistics: Picturing the World (7th E…StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSONThe Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. FreemanIntroduction to the Practice of StatisticsStatisticsISBN:9781319013387Author:David S. Moore, George P. McCabe, Bruce A. CraigPublisher:W. H. FreemanMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons IncProbability and Statistics for Engineering and th…StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage LearningStatistics for The Behavioral Sciences (MindTap C…StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage LearningElementary Statistics: Picturing the World (7th E…StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSONThe Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. FreemanIntroduction to the Practice of StatisticsStatisticsISBN:9781319013387Author:David S. Moore, George P. McCabe, Bruce A. CraigPublisher:W. H. Freeman