1. A multiple choice test has 12 questions with each question having 3 choices for its
answer. The test maker randomly numbers the three choices A, B, and C.
(a) What is the probability that the correct answer for all 12 questions is choice A (i.e., the
correct answers are A,A,A,A,A,A,A,A,A,A,A,A)?
(b) What is the probability that choice B is never the correct answer for the test?
(c) What is the probability that the correct test answers are (in the exact order):
A,B,C,A,B,C,A,B,C,A,B,C?
(d) If the test taker guesses randomly on each question, what is the probability that s/he will get
0% on this test?

2. For a computer system to function correctly, its three main components (HW, SW,
and user/operator) must each function correctly. The probability that each component works
correctly is:

Component / Probability that it works correctly
Hardware / 0.99999
Software / 0.9999
User/Operator (human) / 0.99
Assume that these 3 components work/fail (statistically) independently of each other.
(a) What is the probability that this computer system will work correctly?
(b) What is the probability that one or more of the computer system components will fail?
(c) Criticize our assumption that the hardware and software components fail independently of
each other.

Do you have a specific problem or do you just want someone to do your homework for you?
I don't mean to be rude but I think it would be better if you submitted your answers and asked for people to give you feed back.
I'll answer the first one, just to get you started.
A) (1/3)^12

Hints: For problem 1, if you look only at a single question of the 12-question test, the probability that question will have "A" as an answer is 1/3 since there are three choices. The probability that the answer is "B" or "C" is 1/3 for each, as well, for the same reason. So, the probability that question 1 has answer "A" is 1/3 (we just established that). That means on average, if the test were printed over and over again, and the answers were randomly reordered after each printing, one out of every three tests printed would have "A" as the answer to problem 1. So, what is the probability that problem 2 has "A" for an answer? It is 1/3. So, if you were to look at only tests where "A" is the answer for problem 1, 1/3 of them would have "A" as the answer for problem 2. So, 1/3 of 1/3 is the number of tests where the answers for both problem 1 and problem 2 is "A". That means you multiply the probabilities. This gets you the answer to (a) that Melody2 offered.

Also for problem 1: parts (a) and (c) are similar and parts (b) and (d) are similar. To solve them, you should use the same logic (in other words, (a) and (c) have the same answer and (b) and (d) have the same answer)

For problem 2: If the computer system works correctly, that means that the hardware works correctly AND the software works correctly AND the user performs his/her job correctly. Use what you know about probabilities that are statistically independent. For part (c), during a hardware malfunction, what do you think the probability that the software will continue working will be? If the software malfunctions, what do you think the probability that the user will be able to use the software correctly will be? If these were truly statistically independent, the answers should be 0.9999 and 0.99 respectively. But, that seems highly unlikely to me.