1. (8 pts) One letter at random is picked from the word GOOGLE and independently one letter at random is picked from the word GROUP. a. What is the probability that both letters are 'O'? b. What is the probability that the two letters are the same? Let (Q, F,P) be a probability space and let (B, t0) be a standard Brownian motion. 1. Let t0 (a) Prove that {, t0} is a Brownian motion. (b) Is the stochastic process Y, SB, +2 a Brownian motion? 2. Let 0 s 4. (Conditional Probabilities, Finite Stochastic Processes; 25pts). Three machines A, B and C produce respectively 48%, 27% and 25% of the total number of items of a factory. The percentages of defective output of these machines are 3%, 4% and 8%, respectively. (i). If an item is selected at random, find the probability that the item is defective. (ii). If an item is selected at random and is found to be defective, find the probability that the item was produced by machine A. 4. (Conditional Probabilities, Finite Stochastic Processes; 25pts). Three machines A, B and C produce respectively 40%, 33% and 27% of the total number of items of a factory. The percentages of defective output of these machines are 3%, 5% and 8%, respectively. (i). If an item is selected at random, find the probability that the item is defective. (ii). If an item is selected at random and is found to be defective, find the probability that the item was produced by machine A.