1 Stencil the flea There is a small flea named Stencil. To his right, there is an endless flat plateau. One inch to his left is the Cliff of Doom, which drops to a raging sea filled with flea-eating monsters Cliff of doom

1 Conditional Expectation and Total Expectation There are conditional expectations, just as there are conditional probabilities. If R is a random variable and e is an event, then the conditional expectation Ex(r e)is defined

Problem 1. A couple decides to have children until they have both a boy and a girl. What is the expected number of children that they'll end up with? Assume that each child is equally likely to be a boy or a girl and genders are mutually independent Solution. There are many ways to solve

Problem 1. The following two parts are not related. Try them, to make sure you un- derstand the jargon of random variables distributions, probability density functions, etc. Ask your TA if you don't understand/remember what some phrase means. (a)Suppose X1, X2, and X3 are three mutually independent random variables, each having the uniform distribution

Problem 1. Suppose that you flip three fair mutually independent coins. Define the fol- lowing events: Let be the event that the first coin is heads. · Let be the event that the second coin is heads. · Let be the event that the third coin is heads

The Law of Total Probability is handy tool for breaking down the computation of a prob- ability into distinct cases. More precisely, suppose we are interested in the probability of an event E: Pr(). Suppose also that the random experiment can evolve in two different ways; that is, two different cases X and X are possible. Suppose also that it is easy to find the probability of each

This is a good approach to questions of the form, What is the probability that ntuition will mislead you, but this formal approach gives the right answer every time 1. Find the sample space. ( Use a tree diagram. 2. Define events of interest. Mark leaves corresponding to these events 3. Determine outcome probabilities (a) Assign edge probabilities

Problem 1. Find closed-form generating functions for the following sequences. Do not concern yourself with issues of convergence

Notes for Recitation 14 Counting Rules Rule 1(Generalized Product Rule). Let be a set of length-k sequences. If there are: n1 possible first entries, n2 possible second entries for each first entry, n3 possible third entries for each combination of first and second entries, etc. then:

Notes for Recitation 15 Problem 1. Learning to count takes practice! (a)In how many different ways can Blockbuster arrange 64 copies of 13 conversations about one thing, 96 copies of L'Auberge Espagnole and 1 copy of Matrix Revolutions on a shelf? What if they are to be arranged in 5 shelves?