Problem 4: Lets say, instead, that the test is an adaptive test; you get to answer more questions based on your previous success. This test is structured like this:

First you have to answer three questions and if you are correct on two of them, you get to answer three more questions. If two of those are correct, then you get three final questions, of which you need to get at least two correct to pass the whole test. The test details are: The first test, T1, has three multiple choice questions with four possible answers each (p=0.25 per question). The second test, T2, has three multiple choice questions with three possible answers each (p=0.33 per question). The final test, T3, has three questions that are true/false (p=0.50 each question) The test questions are formed as follows: The questions are in a language you have never seen: a mixture of Navaho, Swahili, Klingon, and Esperanto. So you have to guess on all of the questions and there are no contextual clues to eliminate any answers. This is the first one: 'Arlogh Qoylu'pu'? Moja: Yel kholgo eeah. Mbili: Floroj kreskas irka mia domo. Pe'el! Tatu: La sandvio estos manota'mo'tlhIngan maH! Nne: 'Adeez'`q eeah. (The professor sits at the front of class with a giant, sadistic grin while the students throw wads of paper at his head.) Using the binomial probability rule, the law of total probability and Bayes theorem:

a) What is the probability of getting two right on each sub-exam? (T1, T2, and T3, separately.)

b) What are your overall chances of passing the entire exam?

c) What are your chances of passing T3 if you first pass T1 and T2?