You may use calculators and tables.
1. 70% of mids who come before an honor board are in academic trouble. 60% of mids who come before an honor board have some history of disciplinary trouble.
45% of mids who come before an honor board have both academic and disciplinary trouble.
(a) What fraction have neither academic nor disciplinary problems?
(b) Given that a mid before an honor board does not have academic problems, what is the probability that he/she does not have disciplinary problems?
(c) Given that a mid before an honor board does not have disciplinary problems, what is the probability that he/she does not have academic problems?
(d) If academic trouble and disciplinary trouble were, in fact, independent, what fraction of mids who come before an honor board would have both kinds of trouble?
2. We can't locate a witness for our case. He might be in the library (probability 65%) or in the gym (probability 30%) or in his room (probability 5%).
The probability of finding him if we look in each location is: 15% in the library, 40% in the gym and 100% in his room.
(a) If we look in all 3 locations, what is the chance of finding our witness?
(b) If we fail to find him, what is the probability that he is in the gym?
3. We need to find one more member for the honor board who doesn't know the accused. We figure that 25% of the Brigade doesn't know the accused.
(a) What is the expected number of mids that we will have to interview before finding our board member?
(b) What is the probability that we will need to interview more than 8 mids before finding our member?
(c) Suppose we actually need 3 more members for the board. What is the probability that we will need to interview exactly 8 mids to fill the 3 slots?
4. Honor cases arrive at a rate of 1 every 2 weeks.
(a) What is the probability of no cases in one week?
(b) What is the probability of more than 2 cases in a single week?
5. The honor system isn't perfect. It makes the wrong decision in 10% of the cases.
(a) If we have 20 cases one semester, what is the probability of being wrong in 5 of them?
(b) In the 8 semesters you will be here, what is the probability that the board will be wrong in more than 20 of
the 160 cases it will hear? (Assume that the Central Limit Theorem applies.)
(c) On average, how many cases will be heard until the next wrong decision? (Include the wrong decision in your count.)
6. A stack of honor cases contains 8 cases of cheating and 5 cases of lying. If we choose 4 cases at random, what is the chance we get exactly 1 case of lying?
7. Suppose the length of an honor hearing has a normal distribution with mean 60 minutes and variance 200 minutes.
(a) What is the chance a hearing will be shorter than 45 minutes?
(b) 90% of all hearings are over within what time?
(c) Suppose we can streamline the process and reduce the mean length. (The variance remains the same, however.)
What would the new mean have to be so that 75% of hearings are over within an hour?
8. If you are called to be a witness, the time you have to wait to testify has an exponential distribution with mean 30 minutes.
(a) What is the probability you will have to wait less than 15 minutes to testify?
(b) 95% of the time, you will be done within what time?
9. Suppose the length of a witness's testimony has an exponential distribution with mean 10 minutes. If the board will hear 4 witnesses,
what is the probability that the total time for testimony will be no longer than an hour?
10. Suppose that QPRs for those appearing before an honor board have a uniform distribution on (1.6, 4.0).
(a) What is the mean QPR?
(b) What is the probability that someone appearing before an honor board is SAT (i.e., QPR > 2.0)?
11. Suppose X has a binomial distribution with N=4, p=0.6 and Y has a geometric distribution with p=0.5. If W=X+Y, find Prob(W=3).