Assignment #3      1. Some dice are rolled. You get $1 for each 6 that shows up;         but, if there are no 6's you lose $1 for each die that was         rolled.         What number of dice makes the game most fair?         Most unfavorable?         Most favorable?      2. There are three coins in a bag; one has two heads, one has         two tails, and one is a fair coin. A coin is chosen randomly         from the bag and tossed. A player sees the top of the coin         and makes a bet on what is on the other side of the coin.         What should the guess be? How much should the bet be?         The second question is the more difficult; consider it         a long term project until you have more experience with         probability theory.         Write ISETL funcs to simulate the game and betting strategies.         Use the funcs provided in the file COINBAG.!I      3. Write an ISETL func to return the expected tosses of a fair         coin until n consecutive heads appears. Justify the correctness         of your answer.      4. Here is an algorithm that was suggested for getting a random         subset of a set. If you collect $1 for each use of the random         func, what is the fair price to pay to use the RandSet func?         The answer will depend on the population size and sampleSize.         Write an ISETL func to give the correct answer.         RandSet := func(sampleSize,population);         $ requires: population is a set;sampleSize a positive integer         $ effects : returns a subset of sampleSize from population         local sample;             sample := {};             while #sample /= sampleSize do                 sample := sample with random(population);             end;             return sample;         end;      5. Write ISETL funcs to return the probabilities that exactly n         cards are drawn, (1) with and (2) without replacement, from a deck         before the first ace is drawn. Use the result (2) to  give an         alternate method of solving problem 5 of Assignment 1 about the         expected number of draws.