Probability

2. "Move to Front" Permutations Consider an alphabet of length N. A move-to-front permutation of the N letters consists of picking one of the letters (randomly or otherwise) and moving it to the front of the list. For example, if the alphabet consists of the three letters A, B, and C, and you start with the permutation ABC, then CAB is the result of one move-to-front (the chosen letter is C), as is ABC (the chosen letter is A), but not CBA. a) As a preliminary, recall that all transition matrices are stochastic; that is, each row sums to 1. Suppose the transition matrix of a finite-state, irreducible, aperiodic chain is doubly stochastic; that is, each of its columns also sums to 1. Explain why the stationary distribution must be uniform. b) A standard deck consists of 52 cards. A "random to front" shuffle is defined as follows: Pick one of the 52 cards uniformly at random and move it to the front of the deck (which you are welcome to think of as the top of the deck, if you prefer). Explain why if you perform this move over and over again, in the long run the deck will become well shuffled; that is, all permutations will be equally likely. [Hint: Set up an appropriate chain and use Part a. You might want to try it out first with an alphabet of just the letters A, B, and C.]

Consider the following survey of customer's purchases at a coffee shop. Suppose all selections are made randomly, with replacement. Rewrite each problem with the proper notation, then find the desired probability. a. Find the probability of selecting a single customer who orders decaf coffee. b. Find the probability of selecting a single customer who orders regular coffee or a pastry. c. Find the probability of selecting a single customer who orders both regular coffee and a pastry. d. Find the probability of selecting a single person who had no pastry, given that they ordered a decaf coffee. e. Find the probability of selecting a single person who order decaf coffee, given that they had a pastry. f. Find the probability of selecting two people (with replacement) who both ordered decaf coffee. g. Find the probability of selecting two people (with replacement) where one ordered a pastry and one did not.

3. A casino owner designs a game in which a regular deck of cards is shuffled and a gambler is allowed to select any card at random. If he selects an ace he wins $10. If he selects a face card (i.e. J, Q, or K) he wins $5. If he selects any other card he wins $1. (a) Assuming the casino owner wants at least to break even, what is the minimum value he should charge the gambler to play this game? (b) What is the standard deviation for the gambler's winnings? (c) What is the probability that the gambler wins more than $1? (d) Suppose a gambler plays the game twice. What is the probability that his total winnings exceed $2?

1-A group of people were asked if they had run a red light in the last year. 368 responded "yes", and 168 responded "no". Find the probability that if a person is chosen at random, they have run a red light in the last year. Give your answer as a fraction or decimal accurate to at least 3 decimal places

6- A poll showed that 47.4% of Americans say they believe that some people see the future in their dreams. What is the probability of randomly selecting someone who does not believe that some people see the future in their dreams. Report the answer as a percent rounded to one decimal place accuracy. You need not enter the "%" symbol.

12- Suppose a jar contains 17 red marbles and 34 blue marbles. If you reach in the jar and pull out 2 marbles at random at the same time, find the probability that both are red.

6. Ten sample values have been collected for a non-negative quantity: 137.3, 110.4, 180.7, 61.5, 116.6, 55.8, 301.1, 62.5, 85.3, 519.9. Fit these values to a probability distribution? Justify the distribution you obtain.

21- In a large population, 71 % of the people have been vaccinated. If 5 people are randomly selected, what is the probability that AT LEAST ONE of them has been vaccinated?

1. A bag contains 9 balls, six being black and three being white. You take three balls out at random. What is the probability that: (a) all of them are white? (b) at least two of them are black?

2. Two dice are rolled simultaneously. What is the probability that the product of the two numbers thrown is (a) greater than or equal to 10? (b) an even number?