15.1 Write some statements to simulate spinning a coin 50 times using 0-1 vectors instead of a forloop. Hints:Generate a vector of 50 random numbers, set up 0-1 vectors to represent the heads and tails, and usedoubleandcharto display them as a string ofHs andTs. 15.2 In a game of Bingo the numbers 1 to 99 are drawn at random from a bag. Write a script to simulate the draw of the numbers (each number can be drawn only once), printing them ten to a line. 15.3 Generate some strings of 80 random alphabetic letters (lowercase only). For fun, see how many real words, if any, you can find in the strings. 15.4 A random number generator can be used to estimate?as follows (such a method is called aMonte Carlomethod). Write a script which generates random points in a square with sides of length 2, say, and which counts what proportion of these points falls inside the circle of unit radius that fits exactly into the square. This proportion will be the ratio of the area of the circle to that of the square. Hence estimate?. (This is not a very efficient method, as you will see from the number of points required to get even a rough approximation.) 15.5 Write a script to simulate the progress of the short-sighted student in Chapter 16 (Markov Processes). Start him at a given intersection, and generate a random number to decide whether he moves toward the internet cafe or home, according to the probabilities in the transition matrix. For each simulated walk, record whether he ends up at home or in the cafe. Repeat a large number of times. The proportion of walks that end up in either place should approach the limiting probabilities computed using the Markov model described in Chapter 16.Hint:If the random number is less than 2/3 he moves toward the cafe (unless he is already at home or in the cafe, in which case that random walk ends), otherwise he moves toward home. 15.6 The aim of this exercise is to simulate bacteria growth. Suppose that a certain type of bacteria divides or dies according to the following assumptions: (a) during a fixed time interval, called ageneration, a single bacterium divides into two identical replicas with probabilityp; (b) if it does not divide during that interval, it dies; (c) the offspring (called daughters) will divide or die during the next generation, independently of the past history (there may well be no offspring, in which case the colony becomes extinct). Start with a single individual and write a script which simulates a number of generations. Takep=0.75. The number of generations which you can simulate will depend on your computer system. Carry out a large number (e.g. 100) of such simulations. The probability of ultimate extinction,p(E), may be estimated as the proportion of simulations that end in extinction. You can also estimate the mean size of thenth generation from a large number of simulations. Compare your estimate with the theoretical mean of (2p) n . Statistical theory shows that the expected value of the extinction probability p(E) is the smaller of 1, and (1??p)/p.Soforp=0.75,p(E) is expected to be 1/3. But forp??0.5,p(E) is expected to be 1, which means that extinction is certain (a rather unexpected result). You can use your script to test this theory by running it for different values ofp, and estimatingp(E)in each case.

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