Directions: Each problem from the Module Problem Set will be worked out in its entirety. Your solution method may vary and will not be used against you if you get the right answer. 1. {Bayes' Theorem) Bowl 1 contains six red chips and four blue chips. Five of these 1D chips are selected at random and without replacement and put in Bowl 2, which was originallyr empty. Clue chip is then drawn at random from Bowl 2. Given that this chip is blueT nd the conditional probability that two red chips and three blue chips are transferred from Bowl 1 to Bowl 2. 2. {Conditional Probability) A box contains three coins: two regular coins and one fake two-headed coin {P(H}=l) - You pick a coin at random and toss it. What is the probability that it lands heads up? a You pick a coin at random and toss it, and get heads. 1What is the probability that it is the tum-headed coin? 3. (Conditional Probability and Bayes') A box contains two coins: a regular coin and one fake two-headed coin (P(H)=1). I choose a coin at random and toss it twice. Define the following events. Let A = Event the first coin toss is H, B = Event the second coin toss is H, C = Event the regular has been chosen. We will call that C,. Find P(A|C), P(B|C), P(An BIC), P(A), P(B), andP(An B). Note that A and B are NOT independent, but they are conditionally independent given C. 4. (Combining Multiple Evidence) Consider the following problem of spam filters. The following table holds the joint probability distribution. We can state the following using Bayes' Rule to find probabilities. Table 1: Joint Probability table of Spam Filters. Ham Spam |Total All messages 100 600 1000 With 'free' 100 300 400 With 'viagra' 10 90 100 P(spam token) = P(spam) P(token| spam P(token 600/1000 x 300/600 0.6 x 0.5 P(spam| free) = = 0.75 400/1000 0.4 (1) P(spam|viagra) = 600/1000 x 90/600 0.6 x 0.15 = 0.90 100/1000 0.1 Given these prior probabilities that we can compute and assuming that the words "free" and "Viagra" are independent when we know whether the message is spam (i.e., they are conditionally independent on spam), use the formula for combining multiple evidence to compute P(spam| free nviagra)