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 estimateπas 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 estimateπ. (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.