This considers the sensitive subject of how to ask an embarrasing
question so no one's privacy is violated (a yes/no question). One
can do this by introducing probability into the picture: Have a
fair coin, and follow this procedure (and no one apart from 
participant sees the results of the coin toss) - flip the coin once,
and if it is tails answer the question honestly; if it is heads,
flip the coin again and answer the question according to the coin.
We can test this by Monte Carlo method using Nahin's formula or his
numbers presented in the book.

Associated file: embarrassing.cpp.

The duel between two contestants takes two forms. The first form
is simple alternation of a single shot (first one takes a shot, then
the other). The objective is to find the probability of the first
duellist surviving. The other form is the same except for incrementing
the number of shots a duellist has on each turn, i.e., A takes one shot,
B takes two shots, A takes three shots, etc. until the duel is finished.
We have the same objective as before.

Associated files: duel1.cpp, duel2.cpp, duel3.cpp

A sequence of switches is arranged in three layers so that the
top layer is a serially connected sequence of N switches, comprised 
of a middle layer of N switches connected in parallel each of which
is comprised of a low layer of N switches connected in series. Note
that each of layers has the same number of switches. A series of
switches is considered closed when all of the switches making up the
series are closed. Switches in parallel are closed if any of the
switches are closed. We are to determine the probability that the
light bulb glows (i.e., the circuit comprised of the above switches
is closed) given the number of switches and the individual probability
of single switch being closed (which applies to all the individual
switches in the layered arrangement).

Associated files: glow.cpp, glow1.cpp, and glow2.cpp.

Given that the stronger team has a probability p, (p > .5), of winning,
what is the probability that the weaker team wins the seven game series?
Also, give the probability profile of the series length (for 4 through 7).

Associated file: baseball.cpp.

Actually three different methods of approximating e and one method of
approximating pi. 

1) Divide the unit interval into A equal-sized intervals (in snow1.cpp,
200,000 is selected for A).  Then randomly generate A numbers, fitting them
into the appropriate interval based on the value being within the interval.
Let I(0) be the count of intervals containing 0 of the random numbers.
The ratio A/I(0) will approximate the value of e.
snow1_1.cpp speeds up the process by randomly selecting 1,000,000 numbers
from [0,999999] and incrementing the count according to the number chosen.

2) From the unit interval, count a sequence of randomly selected real numbers
till their sum exceeds 1. A large number of such sequences will find that the
count approximately averages to e.

3) Subdivide the unit interval by the following method: pick a random number from
the unit interval and let that form the right edge of the left subinterval; then
repeat the process substituting the new left subinterval for the unit interval.
Obviously each subsequent left subinterval becomes smaller than the previous
left subinterval. Do this about 10 or 20 times, then take the sum of subsequent
right edges of these left subintervals. Plotting the CDF of this sum, observe
that it is approximately linear in the unit interval. In the text is given that
the slope is equal to e^(-gamma) (Euler's number followed by Euler's constant).

4) Inscribe a circle centered at the origin within a 2x2 square (the radius of
the circle is 1). Then pick points at random within the quarter square that
contains the unit interval for the x and y axes. Let S represent the total count
of random points.  Let C represent the count of the of the subset of points that
also fall within the circle (this section of the circle has area pi/4).
The value C/S * 4 will approximate pi.

Associated files: snow1.cpp, snow1_1.cpp, snow2.cpp, snow3.cpp, and snow_pi.cpp.

N people are flipping independent fair coins n times each. What is the probability
that they all have the same number of heads come up?
Calculate for N = 2, 3, 4 and n = 10, 50, 100, 150.
The title of the chapter is due to the "overwhelming" amount of number-crunching
required to do the problem without a computer.

Associated files: sameflips.cpp

Given that a baseball has a probability p of winning a single game (a simplifying
assumption for sure), what is the probability of winning 81p games after half a
season (assuming a 162 game season)? The same question applies for winning 162p
games for a full season. For p = .01, .02, .., 1.0, graph the ratio of
81p / 162p, explaining the unexpected strangeness of the graph.

Associated files: bbseason.cpp

Initially, one urn contains all black balls, the other urn contains all white
balls.  Then at subsequent time intervals (labeled i = 1, 2, ...), a ball is
selected randomly from each urn and placed in the other urn (in other words
"exchanged").  Obviously at i = 1, this results in one black ball being placed
in the previously all white urn and vice versa.  For subsequent values of i,
the color of ball is dependent on which ball is randomly selected from the
urn.  The objective is to graph the number of black balls as function of i.
How rapidly does the proportion of black balls in urn 1 approach 0.5 as a
function of N, the total number of black balls in urn 1 at i = 0?  Also, how
does the proportion vary as a function of N?

Associated files: ball.cpp

This a form of odd-man out, i.e., N people each simultaneously flip a
a coin and if there is single head or tail among them, that person
has to pay for coffee for all. How many simultaneous flip events on
average are required to determine the payer? A further question is
supposing there is one person with an unfair coin: how does this affect
the outcome?

Associated files: coffee.cpp

What a difference a draw makes! What is the probability that a chess
champ wins a 6 game match? Or ties it? This is oddly difficult for
an analytical solution even though it seems that it should be simple.

Associated files: champ.cpp

This is a needle and circular table top problem: randomly dropping a
needle onto the table top, what is the probability that one end of the
needle sticks out over the edge of the table top? Also consider the
other two events: that of two ends protruding and that of the needle
being completely contained by the interior of the table top.

Associated files: needle.cpp

What is the PDF of Z = X / (X - Y) where X and Y are uniformly distributed
random variables on the unit interval?

Associated files: negativity.cpp

What is the PDF of Z = X ^ Y where X and Y are uniformly distributed
random variables on the unit interval?

Associated files: power.cpp

A capacitor discharging in the presence of resistance and an inductor
will either decay monotonically or with oscillations (an RLC circuit).
The characteristic (quadratic) equation associated with the second
order differential equation of the RLC circuit has a transient response
that decays according to whether the discriminant is positive or
negative. It decays monotonically if it is positive, it oscillates if
negative.

Consider the partially random quadratic formula x^2 + Bx + C = 0. This
is a monic polynomial (leading coefficient is one). What is the
probability, if B and C are uniformly, and independently, randomly
chosen from the unit interval, that the discriminant is positive?

Now consider the same question for the totally random quadratic formula:
Ax^2 + Bx + C =0. Divide through by A and then consider the discriminant
(i.e., is (B/A)^2 > 4(C/A)?).

Associated file: osc_limit.cpp.

Given a probability p of having a boy (1 - p = q is the corresponding
probability of having a girl), how many children are expected till a
subsequent child matches the sex of first? What does the graph look
like for p in the range [0, 0.01, ..., 1.0]?

Associated file: inconceivable.cpp

A radio distress signal was received from a ship that it was near one
of two islands (North and South). N rescue ships are sent,
each with an individual probability s of finding the ship. Note that s
is fixed and the search is independent for each ship. The problem
is that of allocating a number of ships to each island as a function
of the probability that the distressed ship is at that island. For
[0, 0.01, 0.02, ..., 1.0] being the probability of the distressed ship
being at the North island, determine the allocation that maximizes the
probability the distressed ship will be found. Calculate for N = 13, s
= 0.2; N = 40, s = 0.2; N = 40, s = 0.5.

Associated file: tub.cpp

Don't know where the title came from on this one.
Consider the unit square with one side placed along the x-axis, aligned
with the [0, 1] segment.
Select two random real numbers, one from the range [0, 1], the other
from the range [0, pi]. Label the first random number 'x' and the second
'ang'.  Locate x on the x-axis (along one side of the unit square) and then
take the line segment at an angle of ang at x, terminating where it intersects
with the other edge of the unit square. The length of the resulting segment
is the item of interest. Generate 100,000 of these random pairs (x, ang) and
form a histogram of the counts of the segments lengths (show the PDF and CDF).

Associated file: garden.cpp

Two flies land at random on a unit square.  What is the probability that
they are greater that 1 unit apart? The same question is also asked for
the circle of area 1 (hence radius of 1/sqrt(pi)).

Associated files: fly_sqr.cpp, fly_crcl.cpp

A blind spider has a strange kind of web represented by a set of states
and transitions (specifically 8 states or "places" on the web). A fly lands
on the web becoming stuck; it dies for some reason, so the spider has to randomly
search for the fly.  Given that the fly is in state 1 and the spider starts
in state 0, and given the following set of transitions, what is the expected
number of transitions that the spider will make till it reaches the fly?
State 0 transitions to 3, 6, 7 (spider begins search here)
State 1 transitions to 2, 8 (fly is here)
State 2 transitions to 1, 4, 7
State 3 transitions to 0
State 4 transitions to 2, 5
State 5 transitions to 4, 6
State 6 transitions to 0, 5, 7
State 7 transitions to 0, 2, 6, 8
State 8 transitions to 1, 7

Associated files: blind_spider.cpp, blind_spider1.cpp, blind_spider_MCS.cpp

This problem deals with the probability of electronic system failure and
schemes with multiple electronic systems that provide redundancy in an
attempt to reduce the chances of failure.
Given is the probability of single system failure by time T:  1 - e^(-aT)
where 'a' is a positive constant characteristic of the system. This
problem compares the performance of one system and three systems (where
majority rule is then observed, if one system fails, the other two still
provide the correct feedback for decision making). Also assume a = 1/100
and compare results for T = [0, 200].

Associated file: reliable.cpp

This problem has to with a probabilistic pert chart (one of my least
favorite topics, due to experience with Microsoft Project).
Given a certain set of tasks and possible times to complete, plot a
critical likelihood for each task. Also plot a histogram of project
completion counts.

Associated file: pert.cpp