Probability Consider a probability experiment with S = {0, 1, 2, 3, 4, 5, 6, 7} and these two events: A = {1, 2, 3}, and B = {1,3, 5, 6, 7} What is the probability that A occurs? What is the probability that B occurs? What is the probability that An Boccurs? What is the probability that A U B occurs? What is the probability that {} occurs? What is the probability that S occurs?
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- The probability of event A is 55%. The probability of event B is 40%. The probability that A happens given that B has happened is 25%. What is the probability that both A and B happen? 95% 70% 65% 10% 15%During the 17th century, gambler de Méré asked Blaise Pascal an explanation for his game losses. His question was solved by Pascal and Fermat, and yielded the foundations of the theory of probability. Gamblers used to bet on the event of getting at least one ace in four rolls of a dice. As a game variation, De Méré proposed to use two die and roll them 24 times with a bet on having at least one double ace. De Méré thought the games were equivalent but he lost consistently. Explain why? Write a simulation program to empirically determine which one dice/two die game is more likely to win.If the probability of an event is 62%, what is the probability of its complement?
- Size of sample space A six-sided dice is rolled, a five-sided dice is rolled and a three-sided dice is rolled. Considering this as a probability experiment, what is the size of the sample space? An Event In the context of the probability experiment just described (a 6-sided, a 5-sided, and a 3-sided dice thrown): Consider the event in which two face- up numbers on the three dice add to 10. What is the size of this set? Save ResetGive an example of a random variable X : {b, c, d, e} → N (Natural Number) with expectation 2, where each of {b, c, d, e} has equal probabilitySIMULATION AND MODELING Using the mid- square method obtain the random variables using Z0= 1009 until the cycle degenerates to zero.
- Consider the following procedure for initializing the parameters of a neural network: 1. Pick a random number r r = rand(1,1) * (2 + INIT_EPSILON ) – INIT_EPSILON 2. Set e =r for all i, j,l Does this work? No, because the procedure fails to break symmetry. O b. Yes, unless we are unlucky and get r = 0 (up to numerical precision). O. Yes or no, depending on the training set inputs x(i). d. Yes, because the parameters are chosen randomly.Inference question in discrete mathematics. Use the example solution in the picture to solve this inference question. Given the premises: If the steak is well done, it’s overcooked. If the steak is overcooked, the fire alarm will go off. Either the batteries have been changed or the fire alarm will not go off. The batteries have only been changed if the ladder is in the room. The ladder is not in the room. Conclude that the steak is not well done. These premises don’t make sense in the real world. Why? How can you change the premises so that they make sense in the real world? (They don’t have to end up perfectly correct, but they need to avoid the obvious problems you uncover in part B.)Probability and Statistics Consider the following experiment. You draw a square, of width 1 foot, on the floor. Inside the square, you inscribe a circle of diameter 1 foot. The circle will just fit inside the square. You then throw a dart at the square in such a way that it is equally likely to fall on any point of the square. What is the probability that the dart falls inside the circle? (Think about area!)How might this process be used to estimate the value of π?
- For (∃ x)(P(x,b)) Would an example of this being true if the domain was all the Avengers and x was green skin, then "b" being the Hulk would make this true. Am example of this being false would be: If the domain was all integers and x was positive, even integers and "b" was integers greater than zero.Sum of Squared Errors: Remember from your statistics courses that if two random variables X and Y are related by a relation YaX+b and you had a set of observations {(1, 1), (2, 2)..... (z.)). then for every estimated values of a and b, the Sum of Squared Errors (SSE) was defined as in the following formula. SSE(-ar-b)². 1-1 The less SSE, the better a and b are estimated. Consider the following fixed list. In this list each sub-list of length two is standing for one pair (z.). Write a function that recieves two numbers a and b and returns the associated SSE. L- [[1, 2], [1.1, 2], [2, 7.1), (2.5, 7.21, (3, 11]]Electronic Spreadsheet Applications Compare What-If Analysis using Trial and Error and Goal Seek to the given scenario: Let's say a student is enrolled in an online class at a learning institution for a semester. His overall average grade stands at 43% in the course (Term Grade is 45%, Midterm Grade is 65%, Class Participation is 62% and Final Exam is 0%). Unfortunately, he missed his Final Exam and was given 0%. However, he has the opportunity to redo his Final Exam and needs at least an overall average of 60% to pass the course. How can you use Trial and Error and Goal Seek to find out what is the lowest grade he needs on the Final Exam to pass the class? Which method worked best for you and why?