[...]... written explicitly for tossing ve dice, and you may want to modify it to answer similar question for any number of dice, and an arbitrary k-of-a-kind question Exercise 1.2 Run and time YAHTZEE.BAS Then estimate how long a similar problem would run if the question involved tossing 15 fair dice The answer depends on your computer, and the software Both Pascal and C-programs seem to run on my computer about... events, and their chances For the readers who are familiar with elementary probability it may be refreshing to see the computer used for counting elementary events, and randomization used to solve a deterministic optimization problem The questions are What is probability" ? How do we evaluate probabilities in real-life situations? What is the computer good for? 1.1 Mathematical models, and stochastic. .. years! Exercise 1.2 shows the power of old-fashioned pencil -and- paper calculation Problem 1.3 Continuing Problem 1.1, suppose now n identical dice are tossed What is 5n the probability of n , 1 of a kind? ANS: 6n,1 1.3 ELEMENTARY PROBABILITY MODELS 5 1.2.2 Geometric Probability For bounded subsets IRd , put PrA = jjAjj : 1:3 This captures the intuition that the probability of hitting a target is proportional... one-dimensional maximization of a function f x, we would do the following steps 1 Pick an initial best-so-far" point x0 and compute initial value y0 = f x0  2 Select at random x1 in the neighborhood" of x0 and compute y1 = f x1 3 Compare y0; y1 a If y1 y2 then repeat Step 2 b If y1  y0 then make x1 the new best-so-far" y0 := y1; x0 := x1 Then repeat Step 2 4 Stop the program at user request, or when no... to ImproveBest as a replacement for GetPermutation 1.4.5 Random permutations Program RANDTOUR.BAS selects permutations at random only in its simplest" variant Here are a few examples of problems that require selecting random permutations Card games: Poker hand: Select 5 cards at random from a deck of cards Poker 2 players: Select 10 cards at random from a deck of cards Bridge: Split 52 cards into four... white sh and a 100 black sh, and a nearby pond contains 20 black sh and 10 white ones No other sh live there A sh is selected at random from the lake and moved to the pond Then a sh is selected from the pond and moved back to the lake What is the probability that all sh in the pond are black? 1.5.2 Sequential experiments Often the main experiment consist of a sequence of sub-experiments, each depending... Find the probability that heads appears for the rst time on the fourth toss Problem 1.16 from the program RANDTOUR.BAS selects numbers between 1 and n at random until it nds a number not yet on the list Then it ads the number to its list, and repeats the process SUB GetPermutation 1 What is the probability that the second number added to the list required more than k attempts? 2 What is the probability. .. measurable the functions ! IRd In the vector case we also refer to X = X1; : : : Xd as the d-variate, or multivariate, random variable We will use the ordinary notation for sums and inequalities between random variables There is however a word of caution In probability theory, equalities and inequalities between random variables are interpreted almost surely For instance X  Y + 1 means PrX  Y + 1... events S1 ; S2; : : : Sk ; : : : that are independent and have the same probability p = PrSj  We denote by Fj = Sj0 the failure in the j -th experiment, and put q = 1 , p Two important random variables are associated with the binomial experiment are the number X of successes in n trials, and the number T of trials until rst success Example 1.10 The probability that number X of successes in n trials... a random variable X such that f is its probability mass function In modeling random phenomena we can therefore avoid the di culties of designing appropriate sample spaces, and pick directly relevant densities The question, if a density does describe the actual outcomes of experiment is to some extend the question of statistics Properties of various distributions, like lack-of-memory come also handy