ll show some construction defects are 5% for A, 7% for B, and 15% for C. What is the probability for any car bought at random to show some construction defect? 1.5 (M): A bag contains ten billiard balls numbered from one to ten. If two balls are picked at random from the bag, what is the probability of getting: (a) Two balls with even numbers? (b) At least one ball with an odd number? (c) Ball number eight in the selection? You must propose two different methods of solving the exercise. 7 The summing rule can be generalized, assuming X={x1, x2, XN} and Y = {y1, Y2 yM}, into the following: p(yi) =p(yilx1)p(x1)+p(yilx2)p(x2)+ p(yilxn)p(x)) which can also be expressed as a matrix-vector relation, Py = UPx with Px = {p(x1),..., p(xn)} Py={p(y1) p(yM)} being column vectors, and U the M x N "conditional" or "transition" matrix: p (y1|x1) p (y1|x2) p(y1|xn) p (y2|x1) p (yM|x1) p(yM|xN) 1.4 Exercises 1.6 (M): The American roulette has 36 spots numbered 1, 2, 36, which are alterna- tively divided into 18 red spots (odd numbers) and 18 black spots (even numbers), plus two green spots, called 0 and 00. The roulette's outcome is any single number. It is possible to bet on any number (single or combination), or on red, black, odd, even, the first 12 (numbers 1-12), the second 12 (numbers 13-24), and the third 12 (numbers 25-36). The roulette payoffs are 35 to 1 for a winning number, 2 to 1 for a winning red/black/even/odd number and 3 to 1 for a winning first/second/third 12 number. What are the odds of winning any of these bets? Comments? What is the probability of winning at least once after playing single numbers 36 times successively? 1.7 (T): In the lotto game of Exercise 1.3, what are the probabilities of having either exactly one or exactly two winning numbers in the six-number combination? 1.8 (T): This is a television game show where the contestant may win a car. The car is standing behind one of three doors. The payer first designates one of the three doors. The host then opens another door, which reveals a goat! Th