Two infinitely long parallel lines are a distance a apart, and a needle of length ! (! < a) is dropped and spun so as to fall with its midpoint equally likely to lie at any point between the lines, and equally likely to point in any direction. Show that the probability that the needle 21
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- A coin with radius = 1 in is tossed inside a board with floor dimensions of 10 in x12 in. Assume that the coin will land flat and that its landing location will be equally likely among all physically possible positions on the board meaning the coin will always be on the board. What is the probability that the coin's distance from the nearest boundary will be less than 1 in?.Two points are randomly selected on the interval (0, a), where a > 0. It is known that the distance between these two points is less than one-third of the length of the given interval. Determine the probability that both points are in the first half of the interval.. A twig of length 1 is broken into two pieces at random, and one piece is thrown away. Theremaining piece is then also broken at random, and one piece of that is thrown away. Let X be thelength of the twig that remains. What is the probability that X > 1/2?
- A bin of 5 transistors is known to contain 2 that are defective. The transistors are to be tested, one at a time, until the defective ones are identified. Denote by N1 the number of tests made until the first defective is identified and by N2 the number of additional tests until the second defective is identified. Find the joint probability mass func- tion of N1 and N2.Compute the probability that a Random Walk ever returns to the origin and the expected time to the first return.According to some law in genetics, when certain types of peas are crossed, the probability that the plant yields a yellow pea is 0.75 and that it yields a green pea is 0.25. For a plant yielding 90 peas, find the approximate probability that at most 64 peas are yellow. Use the normal approximation with continuity correction.
- A horizontal plane is ruled with parallel lines 2 inches apart. A two-inch needle is tossed randomly onto the plane. The probability that the needle will touch a line is P as attached, where is the acute angle between the needle and any one of the parallel lines. Find this probability.Two drunk start out together at the origin, each having the equal probability of making a step to the left or right along the x-axis. Assume that men make their steps simultaneously. Let N be the total number of steps taken by each man. Assume that a particular instant, they are separated by some distance. In this situation, in a given particular step. If they move towards each other, the separation between them decreases. If the move away from each other, the separation between them increases If they move in the same direction, the separation remains unaltered Then calculate (a) The probability of decreasing their separation (b) The probability of increasing their separation (c) The probability of no separation between them Out of N steps, n1 steps belong to decreasing the separation of them; n2 steps belong to increasing the separation of them and n3 steps belong to no separation. Then, calculate the following. (d) The probability of taking the steps n1, n2 and n3 (e) The…Two drunk start out together at the origin, each having the equal probability of making a step to the left or right along the x-axis. Assume that men make their steps simultaneously. Let N be the total number of steps taken by each man. Assume that a particular instant, they are separated by some distance. In this situation, in a given particular step. If they move towards each other, the separation between them decreases. If the move away from each other, the separation between them increases If they move in the same direction, the separation remains unaltered Out of N steps, n1 steps belong to decreasing the separation of them; n2 steps belong to increasing the separation of them and n3 steps belong to no separation. Then, calculate the following. (a) The probability of taking the steps n1, n2 and n3 (b) The probability that they meet again at N steps
- . A frog is performing a random jump on a Pentagon(5 sides) starting from corner 1. For each jump, it jumps with equal probability to one of the two adjacent corner with equal likelihood. The frog repeats this process until it returns to the corner where it began. When this ends, what is the probability that the frog has jumped on every corner of the pentagon? How would you generalize this case for a frog jumping on the corners of an n-sided polygonThree distinct integers are chosen at random from the first 20 positiveintegers. Compute the probability that: (a) their sum is even; (b) their product iseven.A jury of 12 people is selected at random from a group of 16 men and 18 women. What is the probability that the jury contains exactly 7 women? Suppose that the jury is selected at random from a group of 1600 men and 1800 women. Use the binomial approximation to the hypergeometric distribution to calculate the probability that in this case the jury contains exactly 7 women