Consider an urn initially containing n balls, numbered 1 through n, and suppose that balls will be randomly drawn from the urn, one by one, and without replacement (so that after n draws, it is empty). Letting X be the number of successes that will occur, where a success is considered to occur on the ith draw if the ball obtained is numbered i or smaller, give the expected value of X. (E. g., if n = 5, and the balls are drawn in the order 3, 1, 5, 4, 2, then x = 3, because the 2nd, 4th, and 5th draws result in successes, but the 1st and 3rd draws don’t.)

If f(x)=x^3 and g(x)=2x+7 when g(x)=-5

Tamar is measuring the sides and angles of tuv to determine whether it is congruent to the triangle below. which pair of measurements would eliminate the possibility that the triangles are congruent

Amachine that produces a special type of transistor (a component of computers) has a 2% defective rate. the production is considered a random process where each transistor is independent of the others. (a) what is the probability that the 10th transistor produced is the first with a defect? (b) what is the probability that the machine produces no defective transistors in a batch of 100? (c) on average, how many transistors would you expect to be produced before the first with a defect? what is the standard deviation? (d) another machine that also produces transistors has a 5% defective rate where each transistor is produced independent of the others. on average how many transistors would you expect to be produced with this machine before the first with a defect? what is the standard deviation? (e) based on your answers to parts (c) and (d), how does increasing the probability of an event a↵ect the mean and standard deviation of the wait time until success?