“Solve” the Monty Hall problem in the case of 4 doors (3 goats, 1 car, host opens 1 door after your initial selection), using Bayes’ Rule. Assume a uniform prior distribution; provide sufficient detail and argue why it is either advantageous to stay or switch your initial selection following the reveal.

. To save time Nivaldo's chemistry professor grades only 3 randomly chosen problems of a 12-problem assignment. Suppose that Nivaldo has 9 out of 12 problems correct on the assignment. a) Compute the probability that the 3 chosen problems are correct (Nivaldo scores 100%). b) Compute the probability that all 3 chosen problems are incorrect (Nivaldo scores 0%). c) Compute the expected value of the number of correct problems chosen by the professor.

Three randomly selected households are surveyed. The numbers of people in the households are 1, 3, and 8. (a) Construct a table representing the sampling distribution of the sample proportion of households with even numbers of people when samples of sizes n = 2 are randomly selected. You should follow this process: List all possible samples (with replacement!) of size 2. (Hint: There are 9 possible samples). Find the proportion of even numbers in each sample. Construct the probability distribution by listing each different sample proportion and its probability or frequency (b) Find the mean of the sampling distribution of the sample proportion of households with an even number of people. (c) Find the actual proportion of households with even numbers (from the three households sampled).(d) Based on your results, is the sample proportion an unbiased estimator of the population proportion? Why or why not?

A Social Science Statistics classroom contains 13 Psychology majors, 9 Sociology majors, 4 business majors, 5 nursing majors, and 4 students with other majors.  Mayra is sampling WITH REPLACEMENT from this classroom of students.  For the first student that she selects, the probability of selecting a Psychology major is about ____.  For the second student that she selects, the probability of selecting a Psychology major is about ____.  For the third student she selects, the probability of selecting a Psychology major is about ____.