The probability distribution for the random variable x follows . X F ( X ) 20 0 . 21 24 0. 16 31 0 . 28 37 0 . 35 a . Is this a valid probability distribution ?" Select $ b . What is the probability that * = 31 ( to 2 decimals ) ? C . What is the probability that * is less than or equal to 24 ( to 2 decimals ) ? d . What is the probability that x is greater than 31 ( to 2 decimals ) ?Suppose that we have a sample space S = {E1, E2, E3, E4, E5, E5, E7}, where E1, E2, ..., E7 denote the sample points. The following probability assignments apply: P(E1) = 0.05, P(Ez) = 0.2, P(E3) = 0.15, P(E4) = 0.25, 19035) = 0.15, P(E5) = 0.1, and P(E7) = 0.1. Assume the following events when answering the questions. A = {Evil-Es} 3\" {52-51051} C= {Ez-EaeESsE'Jl a. Find P(A), P(B), and P(C). P(A) P(B) P(C) b. What is P(A U B)? c. What is P(A 0 B)? d. Are events A and C mutually exclusive? e. What is P(BC )? Many students accumulate debt by the time they graduate from college. Shown in the following table is the percentage of graduates with debt and the average amount of debt for these graduates at four universities and four liberal arts colleges. University %with Debt Amount\") College %with Debt Amount\") 1 78 32,930 1 88 28,754 2 67 32,110 2 99 29,000 3 55 11,227 3 53 10,205 4 62 11,853 4 46 11,012 a. If you randomly choose a graduate of College 2, what is the probability that this individual graduated with debt (to 2 decimals)? b. If you randomly choose one of these eight institutions for a follow-up study on student loans, what is the probability that you will choose an institution with more than 50% of its graduates having debt (to 3 decimals)? c. If you randomly choose one of these eight institutions for a followup study on student loans, what is the probability that you will choose an institution whose graduates with debts have an average debt of more than $ 20,000 (to 3 decimals)? cl. What is the probability that a graduate of University 1 does not have debt (to 2 decimals)? e. For graduates of University 1 with debt, the average amount of debt is $ 32,930. Considering all graduates from University 1, what is the average debt per graduate? Round to nearest dollar. $