Solve both questions hand written and show work

1. Three transmission shaft factories, represented by a random variable X, and the quality of the transmission shaft represented by a random variable Y, are under consideration. X can be 1, 2, or 3 (1 representing factory 1, 2 representing factory 2, and 3 representing factory 3). A transmission shaft can be defective or good, and Y can be considered to be 1, or 2. Suppose that 100 transmission shafts are evaluated and the information given in the following table is obtained. 2 3 1 (Defective) 10 20 25 2 (Good 20 15 10 (a) Find the joint PMF of X and Y (5 points) (b) Find the marginal PMF of X and Y (5 points) (c) If Y=2, find the conditional PMF of A7 (5 points) . Suppose that 3 balls are chosen without replacement from an urn consisting of 5 white and 8 red balls. Let X, equals 1 if the i-th ball selected is white and let it equals O otherwise. Give the joint probability mass function of (a) X1, X2; (10 points) (b) X1, X2, X3: (10 points)