Probability

ICEF Introductory Probability and Statistics Home assignment 3 21. 09. 2020 Submit solutions to your class teachers before the class on the week beginning 28.09.2020. Problem 1. A princess is looking for a knight who will marry her. With probability % the next knight she'll meet will be brave, with probability % he will be rich and with probability % both brave and rich. (a) What is the probability he will be neither rich nor brave? (b) What is the probability he will be brave given that he is not rich? (c) What is the probability he will be brave given that he is rich or brave? (Be careful, its's not 1.) Problem 2. A box contains a white and m black balls. One ball is added randomly, white with probability p and black with probability 1 p. Then a random ball is taken out from the box. (a) Find the probability that the taken ball is white. (b) It turned out that the taken ball is white. Find the conditional probability that the added ball was white. Problem 3. To reduce theft, a company proposes to screen the workers with a lie-detector test that has been proved correct 80% of the time for guilty subjects and 90% correct for innocent subjects. The company will re all the workers who fail the test. Suppose that 5% of workers steal from time to time. (a) If a worker is red, what is the conditional probability he is innocent? (b) If a worker is not red, what is the conditional probability he is guilty? Problem 4. An ofce worker has three ties of different colors: red, blue, and yellow. Every working day he randomly decides which one of them to wear. (a) What is the probability that during a week he will never wear the yellow tie? Assume that there are 5 working days in a week. (b) A colleague of the worker is sure that she saw him wearing the red tie at least twice during the last 5 days. Given this, what is the probability that the worker wore the red tie exactly 3 times during the last 5 days? Problem 5. The managers of a car store observe that 70% of all the cars they sell are bought by people who already own a car. They also observe that 50% of the customers who come to the store but do not buy a car, already own one. Furthermore, 40% of the customers coming to the store buy a car. (a) What is the probability that a person coming to the shore already owns a car? (b) What is the probability that a car will be sold to a person who already has one? (c) What is more dicult for the managers: to sell a car to a person who already owns a car or to a person who does not own one