Question 1: How can two cubical (six sided) dice be labeled using numbers {0,1,2,3,4,5,6} so that when the two dice are thrown, the sum has a uniform probability distribution over the integers ( 1,2...12}.The same number can repeat more than once on a face of either die, but each face will come up with the same probability 1/6 when the die is rolled.

Question 2: A test for a disease is 90% reliable, i.e. in 90% of cases of people who have the disease, a positive test result is returned and in 90% of the people who do not have the disease, a negative result is obtained. Suppose that 1% of people are known to have the disease. If a person X takes the test and gets a positive result for the test, what is the probability that X actually has the disease?

Question 3: Alice and Bill flip a fair coin until one of the respective patterns A = HHT or B = HTT appears for the first time (and then the corresponding player wins).

Here is a state diagram that indicates how Alice (or Bill) would proceed to a win: the transitions indicate flips of a fair coin (H/T).

Use the state diagram to calculate by hand, the exact probability that Alice wins within 5 or fewer coin flips and via a programmed function that takes n 3 as argument and returns the ratio of the probability that Alice wins in exactly n flips to the probability that Bill wins in exactly n flips.

Question 4: Redo problem 3 but this time choose winning patterns A = THH and B = HHT. You will need to draw a state diagram based on these patterns before answering parts (a) and (b) as above.

Be sure to include comments. The comment should describe the purpose of the program and the data to be entered.