(5%) Given: The probabilities of three events, A, B, and C, occurring areP(A) = 0.35, P(B) = 0.45, and P(C) = 0.2. Assuming that A, B, or C hasoccurred, the probabilities of another event, X, occurring are P(X│A)=0.8,P(X│B)=0.65, and P(X│C)=0.3. Find P(A│X)= _____________.2. (5%) A Ph.D. graduate has applied for a job with two universities: A and B.The graduate feels that she has a 60% chance of receiving an offer fromuniversity A and a 50% chance of receiving an offer from university B. If shereceives an offer from university B, she believes that she has an 80% chanceof receiving an offer from university A. What is the probability that atleast one university will make her an offer?Probability = _________________3. (5%) The monthly sales at a bookstore have a mean of 50,000 and a standarddeviation of 6,000. Profits are calculated by multiplying sales by 40% andsubtracting fixed costs of 12,000. Find the standard deviation of monthly profits.Standard deviation of monthly profits = ____________4. (5%) Suppose a disease is present in 3% of population. A diagnostic test forsuch disease shows 10% false positive and 5% false negative. That is, for apatient having the disease,, the test shows positive (+) with probability0.95 and negative (-) with probability 0.05. For a patient not havingdisease, the test shows positive (+) with probability 0.10 and negative (-)with probability 0.90. If a patient’s test show (+), what’s the probabilityof his having the disease?Probability = ______________5. (6%) Throw a dice n times fairly and observe the number of dice each time.Assume a random variable X is the frequency that the number of dice is one.(3%) Write down the probability distribution of random variableb. (3%) Assume n = 5, P(X=5)=______________6. (5%) A communication system consists of n components, each of which will,independently, function with probability p. The total system will be able tooperate effectively if at least one-half of its components function. For whatvalues of p is a 5-component system more likely to operate effectively than a3-component system?Values of p should be ___________7. (6%) Suppose that earthquake occur in the eastern part of Taiwan inaccordance with the assumptions for the Poisson probability distribution at a