Given a Sample Space S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, Event A = {1, 3, 4, 7, 9}, and Event B = {3, 7, 9, 11, 12, 13} Find the probability P(A|B). State your answer as a value with one digit after the decimal point.
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2.
Given a Sample Space S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13},
Event A = {1, 3, 4, 7, 9}, and
Event B = {3, 7, 9, 11, 12, 13}
Find the probability P(A|B). State your answer as a value with one digit after the decimal point.
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- please try to simulate the probability of rolling a Die with Sample Space* S={1,2,3,4,5,6} and the probability of each sample point has a 1/6 chance of occurring, i.e., you need to verify that your simulation converges to 1/6 when you select one point of sample space. When X is a random variable for sample point of rolling a Die, Pr(X<=4)=2/3. Please verify this result by simulation. Please let me know how to make an Excel file as stated above.Let l be a line in the x-y plane. If l is a vertical line, its equation is x 5a for some real number a. Suppose l is not a vertical line and its slope is m. Then the equation of l is y 5mx 1b, where b is the y-intercept. If l passes through the point (x0, y0,), the equation of l can be written as y 2y0 5m(x 2x0 ). If (x1, y1) and (x2, y2) are two points in the x-y plane and x1 ≠ x2, the slope of line passing through these points is m 5(y2 2y1 )/(x2 2x1 ). Write a program that prompts the user two points in the x-y plane. The program outputs the equation of the line and uses if statements to determine and output whether the line is vertical, horizontal, increasing, or decreasing. If l is a non-vertical line, output its equation in the form y 5mx 1b.Let l be a line in the x-yplane. If l is a vertical line, its equation is x = a for some real number a. Suppose l is not a vertical line and its slope is m. Then the equation of l is y = mx + b, where b is the y-intercept. If l passes through the point (x₀, y₀), the equation of l can be written as y - y₀ = m(x - x₀). If (x₁, y₁) and (x₂, y₂) are two points in the x-y plane and x₁ ≠ x₂, the slope of line passing through these points is m = (y₂ - y₁)/(x₂ - x₁). Instructions Write a program that prompts the user for two points in the x-y plane. Input should be entered in the following order: Input x₁ Input y₁ Input x₂
- Given A = {1,2,3} and B={u,v}, determine. a. A X B b. B X BUsing the law of total probabilty Suppose we have a sample space S and two events A and B such that S = AUB and AnB= 0 and an event E, Suppose we have the following results: P(A) = 1/3 P(B) = 2/3 P (E|A) = 1 P(E|B) = 0 What is P(E) ? Write your answer as a single fraction of integers. For example, if the answer were 17/20 then enter 17/20 with no spaces or other symbols. You may reduce the fraction by cancelling factors common to the numerator and denominator but you do not need to do this.Let pn(x) be the probability of selling the house to the highest bidder when there are n people, and you adopt the Look-Then-Leap algorithm by rejecting the first x people. For all positive integers x and n with x < n, the probability is equal to p(n(x))= x/n (1/x + 1/(x+1) + 1/(x+2) + … + 1/(n-1)) If n = 100, use the formula above to determine the integer x that maximizes the probability n = 100 that p100(x). For this optimal value of x, calculate the probability p100(x). Briefly discuss the significance of this result, explaining why the Optimal Stopping algorithm produces a result whose probability is far more than 1/n = 1/100 = 1%.
- Generate 100 synthetic data points (x,y) as follows: x is uniform over [0,1]10 and y = P10 i=1 i ∗ xi + 0.1 ∗ N(0,1) where N(0,1) is the standard normal distribution. Implement full gradient descent and stochastic gradient descent, and test them on linear regression over the synthetic data points. Subject: Python ProgrammingRemaining Time: 2 hours, 27 minutes, 02 seconds. Duestion Completion Status: Y=A'. B' + (AOB) b. Using Karnaugh map, simplify the following Boolean function F (show your grouping): [2 Marks] A B C F 1 1 1 1 1 1 1 1 1 0. 1 1 1 1 1 1 Click Save and Submit to save and submit. Click Save All Answers to save all answers. Save All A O Type here to search DLL FS F9 F10 F11 9 3 r 7 Y 8 W E R = G i JJ 立Pick one million sets of 12 uniform random numbers between 0 and 1. Sum up the 12 numbers in each set. Make a histogram with these one million sums, picking some reasonable binning. You will find that the mean is (obviously?) 12 times 0.5 = 6. Perhaps more surprising, you will find that the distribution of these sums looks very much Gaussian (a "Bell Curve"). This is an example of the "Central Limit Theorem", which says that the distribution of the sum of many random variables approaches the Gaussian distribution even when the individual variables are not gaussianly distributed. mean Superimpose on the histogram an appropriately normalized Gaussian distribution of 6 and standard deviation o = 1. (Look at the solutions from the week 5 discussion session for some help, if you need it). You will find that this Gaussian works pretty well. Not for credit but for thinking: why o = 1 in this case? (An explanation will come once the solutions are posted).
- You are given a N*N maze with a rat placed at maze[0][0]. Find whether any path exist that rat can follow to reach its destination i.e. maze[N-1][N-1]. Rat can move in any direction ( left, right, up and down).Value of every cell in the maze can either be 0 or 1. Cells with value 0 are blocked means rat cannot enter into those cells and those with value 1 are open.Input FormatLine 1: Integer NNext N Lines: Each line will contain ith row elements (separated by space)Output Format :The output line contains true if any path exists for the rat to reach its destination otherwise print false.Sample Input 1 :31 0 11 0 11 1 1Sample Output 1 :trueSample Input 2 :31 0 11 0 10 1 1Sample Output 2 : false Solution: //// public class Solution { public static boolean ratInAMaze(int maze[][]){ int n = maze.length; int path[][] = new int[n][n]; return solveMaze(maze, 0, 0, path); } public static boolean solveMaze(int[][] maze, int i, int j, int[][] path) {//…We have considered, in the context of a randomized algorithm for 2SAT, a random walk with a completely reflecting boundary at 0—that is, whenever position 0 is reached, with probability 1 we move to position 1 at the next turn. Consider now a random walk with a partially reflecting boundary at 0– whenever position 0 is reached, with probability 1/4 we move to position 1 at the next turn, and with probability 3/4 we stay at 0. Everywhere else the random walk either moves up or down 1, each with probability 1/2. Find the expected number of moves to reach n starting from position i using a random walk with a partially reflecting boundary. (You may assume there is a unique solution to this problem, as a function of n and i; you should prove that your solution satisfies the appropriate recurrence.)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.