l) Probabilistic reasoning {311) You have been given the assignment to diagnose a certain disease D given the two symptoms S and T. All the random variables D, S and T are binary variables, with the domains D = {+d, I:i], S = [+s, s] and T = {+t, t}. A value ofS = +3 means a patient has symptom S, whereas a value of .5' = s means a patient does not have symptom 5. Likewise, the values +t and t relate to a patient having symptom T or not. Lastly, the values +6! and d imply.r a patient has the disease D or not. To get a feeling for the empirical probabilities for the various random variables, you compile a large number of patient journals and enter the resulting data into a database, where each row includes the three attributes D, S and 1". From the database, you obtain the table below, containing the iii joint probobiiiry distribution P{D,.S',T,'I. Using the ll joint probability dism'bun'oni calculate the probabilities below. To get full points. you must use the equations from probability theory in your calculations. showing how you obtained your results. by substituting the values in the table above into your equations. a) What is the marginal probability for D = +d. i_e. p(+d)'? (0.5p) b) What it thejoint probability forS = +3 and T = -t, i.e. p(+S. t)? (0.5p) c) What is the conditional probability for .S' = +5 and T = -t. (Dip) given D = +d, i.e. p(+s, -t| + d)? d) What is the probability that a new patient is diagnosed with the disease (0.5p) i.e. D = +d given that the patient has symptom S but not T, i.e. p(+d| + s, -t}? To get full points, you must use Bayes Rule. e) What probability would you get in (d) if you assume 5 and T are (1p) conditionally independent of each other. given D, i.e. .5' J. TID