There are two roads from A to B and two roads from B to C. Each of the four roads is blocked by snow with probability p, independently of the others. Find the probability that there is an open road from A to B given that there is no open route from A to C. If, in addition, there is a direct road from A to C, this road being blocked with probability p independently of the others, find the required conditional probability.

A company sells a product which is either of low or high quality where the quality is known to the company but not to the buyer, though the buyer learns the product’s quality after purchasing and consuming it. The product is of high quality with probability q, and low quality with probability 1 - q. It is of value v to the buyer when it is high quality and of zero value when it is low quality. After Nature chooses the product’s quality, it is revealed to the seller but not to the buyer. The seller then chooses an introductory price (the price charged for the buyer’s first purchase) which can be any non- negative number. The buyer observes the introductory price and decides whether to buy. If the buyer chooses not to buy then the payoff to both the seller and the buyer is zero. If the product is purchased at an introductory price p and proves to be low quality then the buyer never buys it again and realizes a payoff of -p. If the buyer purchases it and it proves to be high quality then…

An ant is at one vertex of a square and needs to get to the opposite vertex (where there is food), walking along the sides of the square. The problem is that on each side, with probability p, there may be a predator preventing it from passing. The ant tries to go through where there are no predators. It is known that the events related to the presence of the predator on each side They are independent. Find the probability that a) The ant gets its food. b) There is at least one predator in the square, knowing that the ant gets its food. c) There are 4 predators in the square, knowing that the ant did not get its food. Need help with letter b :(

One of the n encrypted files in a folder is a special file. K of these files are removed one at a time by an attacker who has access to the folder. Each choice has an equal chance of being any of the currently available files. What is the probability that the person will withdraw the special file?You must justify or explain how you arrived at your conclusion.