due Tuesday, June 19.

Prepare an Excel sheet where the first column contains the numbers 2, 3, ... up to 50. The second column should display the probability that group of n people (where n is the number in the same row and in the first column) have two persons with the same birthday.

Confirm the theoretical results above for n=22. 10 times compute 22 random numbers between 1 and 365. Sort them to detect same birthdays. What is the empirical probability?

Candidates are asked four questions one after one. There are four levels of difficulty of these questions: easy, medium, difficult, and very difficult. The score the candidate has so far determines what question will be asked:

- If the candidate didn't answer any question wrong so far, a very difficult question is asked. The probability that the candidate can answer such a question is 1/5.
- If the candidate did only answer one question wrongly so far, a difficult question is asked. The probability that the candidate can answer such a question is 1/4.
- If the candidate answered two question wrong so far, a medium question is asked. The probability that the candidate can answer such a question is 1/3.
- Finally, if the candidate was asked three questions so far and answered all of them wrongly, an easy question is asked. The probability that the candidate can answer such a question is 1/2.

There are two candidates. Adam, where the above probabilities apply as described, and Beth, whose performance depends very strongly on her feelings, and the feelings depend on whether she answered the previous question right or wrong. If Beth answered the last question correctly, she is so elevated that her probability to answer the next question correctly is increased by 0.1. Therefore then (but only with this previous correct answer) she will answer easy, medium, difficult, or very difficult questions correctly with probabilities 0.3, 0.35, 0.4333, and 0.6, respectively. If her last answer was wrong, however, she faces the same probabilities than Adam.

Each candidate gets $1000 for every right answer.

Use a tree diagram to answer the following questions:

a) What values can the random variable "Money won" take?

b) Compute the expected value, the standard deviation, and display the probability distribution for the random variable "Money won" both for Adam and Beth.

c) What is the probability that Adam wins at least $3,000? What is the probability that Beth wins $1000 or less?

d) What is the probability that Adam wins more than Beth? What is the probability that Beth wins more than Adam?

Erich Prisner, June 2007