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.]

3. [6] Consider the experiment of throwing two dice. (a) [3] Find the probability of the event that the second dice shows a larger number of dots than the first one. (b) [3] Find the probability of the even that the second dice shows a larger number of dots than the first one given that the sum of the two dice is not greater than 4.

7. [5] Messages arrive at a computer at an average of 15 messages per second. The number of messages that arrive in 1 second is known to be a Poisson random variable. (a) [5] Find the probability that more than 3 messages arrive in 1 second.

9. [10] A computer generates hexadecimal characters (e.g. 0000, 0001, 0010, etc.). Let \X" be the integer value corresponding to a hex character. Suppose that the four binary digits in the character are independent and each is equally likely to be \0" or \1". (a) [2] Describe the underlying space of this random experiment and specify the probabilities of its elementary events. (b) [3] Show the mapping fromS to Sx. (c) [5] Find the probabilities for the various values ox.

3. The executive of the Manitoba Association of Mathematics Teachers consists of 3 women and 2 men. In how many ways can a president and secretary be chosen if: (a) the president must be female and the secretary male? (b) The president must be male and the secretary female? (c) The president and secretary are of opposite sex?

1. Consider the following survey of students at South. Suppose all selections are made randomly, (with replacement). Rewrite each problem with the proper notation, and find the desired probability. a. Find the probability student is in Prof/Tech. b. Find the probability student is in Academic Transfer and Part Time. c. Find the probability student is in Prof/Tech or Full Time. d. Find the probability student is Full Time given that they are in Academic Transfer. e. Find the probability student is Academic Transfer given that they are Part Time. In selecting two students at random, with replacement... f. Find the probability both (first and second) students are Part Time. g. The first is in Academic Transfer and the second is in Prof/Tech

a) add columns /rows to field totals for each category. b) In selecting person at random, what is the probability that they are.... i) P (male smoker) ii) P (smoker) iii) P (Non-Smoker) iv) P (female Non-Sunder

3. Suppose A and B are mutually exclusive, with P(A) = 0.40 and P(B)= 0.25. a. What is the probability of A given B? b. What is the probability of A and B?

Find the probabilities: a) P(rolling two ones) b) P(rolling doubles) c) P (rolling exactly one 4) d) P (Sum is 7) e) P (Both even #'s) f) P (sum is even) 9) P(rolling a 3) h) P (sum is 13)

2. Of fifty students surveyed, twenty-five played volleyball thirty-three played basketball and four played neither sport. a) How many students played both sports? b.) Complete the Venn-Diagram. c) Find the probabilities: i) P ( play both sports) ii) P (play Volleyball only) (iii) P (play neither) iv) P (play Voleyball or basketball)