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<html>
<head>
<title>
HIGH_CARD_SIMULATION - Guessing What the Highest Card Is
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
HIGH_CARD_SIMULATION <br> Guessing What the Highest Card Is
</h1>
<hr>
<p>
<b>HIGH_CARD_SIMULATION</b>
is a C++ program which
simulates a game in which you have one chance to select the highest card
from a deck, using GNUPLOT to display the estimated and exact probabilities.
</p>
<p>
You are given a deck of DECK_SIZE cards.
</p>
<p>
Your goal is to select the high card. For convenience, we can assume
the cards are a permutation of the integers from 1 to DECK_SIZE, but in
fact the user mustn't see such values or else it's obvious which is the
largest card. (For a truly difficult problem, we'd have to work harder
to disguise the range of values and the likely distribution!)
</p>
<p>
Your choice is made under the following rules: You may turn over
one card at a time. When a card is turned over, you may declare that to be
your choice, or else turn over another card. If you have not chosen a card
by the end, then your choice is the final card.
</p>
<p>
If you have no idea what to do, and simply decide in advance to pick
a card "at random", that is, for example, you decide to pick the 15th card
before having seen any cards, then your probability of winning is 1/DECK_SIZE.
</p>
<p>
The question is, can you do better than that?
</p>
<p>
A good strategy is as follows: always look at the first SKIP_NUM cards
without choosing them. Then choose the very next card you encounter
that is larger than the cards you skipped.
</p>
<p>
Using this program, you can easily see that skipping 5 cards is much better
than picking one at random, skipping 10 is even better, and so on...up to some
point, when the benefit begins to disappear. Of course,
you can't skip too many cards, and in fact, the results seem to be best for
somewhere around 30 to 35 cards skipped. For problems like this, the
optimal value is somewhere around 1 / e, where E is the base of the natural
logarithm system.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>HIGH_CARD_SIMULATION</b> is available in
<a href = "../../c_src/high_card_simulation/high_card_simulation.html">a C version</a> and
<a href = "../../cpp_src/high_card_simulation/high_card_simulation.html">a C++ version</a> and
<a href = "../../f_src/high_card_simulation/high_card_simulation.html">a FORTRAN90 version</a> and
<a href = "../../f77_src/high_card_simulation/high_card_simulation.html">a FORTRAN77 version</a> and
<a href = "../../m_src/high_card_simulation/high_card_simulation.html">a MATLAB version.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/brownian_motion_simulation/brownian_motion_simulation.html">
BROWNIAN_MOTION_SIMULATION</a>,
a C++ program which
simulates Brownian motion in an M-dimensional region.
</p>
<p>
<a href = "../../cpp_src/duel_simulation/duel_simulation.html">
DUEL_SIMULATION</a>,
a C++ program which
simulates N repetitions of a duel between two players, each of
whom has a known firing accuracy.
</p>
<p>
<a href = "../../cpp_src/fair_dice_simulation/fair_dice_simulation.html">
FAIR_DICE_SIMULATION</a>,
a C++ program which
simulates N tosses of 2 dice, making a histogram of the results.
</p>
<p>
<a href = "../../cpp_src/gnuplot/gnuplot.html">
GNUPLOT</a>,
C++ programs which illustrate how a program can write data
and command files so that gnuplot can create plots of the program results.
</p>
<p>
<a href = "../../cpp_src/ising_2d_simulation/ising_2d_simulation.html">
ISING_2D_SIMULATION</a>,
a C++ program which
carries out a Monte Carlo simulation of an Ising model,
a 2D array of positive and negative charges,
each of which is likely to "flip" to be in agreement with neighbors.
</p>
<p>
<a href = "../../cpp_src/poisson_simulation/poisson_simulation.html">
POISSON_SIMULATION</a>,
a C++ library which
simulates a Poisson process in which events randomly occur with an
average waiting time of Lambda.
</p>
<p>
<a href = "../../cpp_src/reactor_simulation/reactor_simulation.html">
REACTOR_SIMULATION</a>,
a C++ program which
a simple Monte Carlo simulation of the shielding effect of a slab
of a certain thickness in front of a neutron source. This program was
provided as an example with the book "Numerical Methods and Software."
</p>
<p>
<a href = "../../cpp_src/snakes_and_ladders/snakes_and_ladders.html">
SNAKES_AND_LADDERS</a>,
C++ programs which
simulate the game of Snakes and Ladders.
</p>
<p>
<a href = "../../cpp_src/string_simulation/string_simulation.html">
STRING_SIMULATION</a>,
a C++ program which
simulates the behavior of a vibrating string
by solving the corresponding initial boundary value problem (IBVP),
creating files that can be displayed by gnuplot.
</p>
<p>
<a href = "../../cpp_src/three_body_simulation/three_body_simulation.html">
THREE_BODY_SIMULATION</a>,
a C++ program which
simulates the behavior of three planets, constrained to lie in a plane,
and moving under the influence of gravity,
by Walter Gander and Jiri Hrebicek.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Paul Nahin,<br>
Digital Dice: Computational Solutions to Practical Probability Problems,<br>
Princeton University Press, 2008,<br>
ISBN13: 978-0-691-12698-2,<br>
LC: QA273.25.N34.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "high_card_simulation.cpp">high_card_simulation.cpp</a>, the source code.
</li>
<li>
<a href = "high_card_simulation.hpp">high_card_simulation.hpp</a>, the include file.
</li>
<li>
<a href = "high_card_simulation.sh">high_card_simulation.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "high_card_simulation_prb.cpp">high_card_simulation_prb.cpp</a>
a sample calling program.
</li>
<li>
<a href = "high_card_simulation_prb.sh">high_card_simulation_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "high_card_simulation_prb_output.txt">high_card_simulation_prb_output.txt</a>,
the output file.
</li>
<li>
<a href = "test02_commands.txt">test02_commands.txt</a>,
the GNUPLOT command file;
</li>
<li>
<a href = "test02_data.txt">test02_data.txt</a>,
the data file.
</li>
<li>
<a href = "test02.png">test02.png</a>,
plots the exact chances of winning with various skips, and a deck of 100 cards.
</li>
<li>
<a href = "test03_commands.txt">test03_commands.txt</a>,
the GNUPLOT command file;
</li>
<li>
<a href = "test03_data.txt">test03_data.txt</a>,
the data file.
</li>
<li>
<a href = "test03.png">test03.png</a>,
plots the exact chances of winning with various skips, and a deck of 100 cards.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>HIGH_CARD_PROBABILITY:</b> winning probabilities for the high card game.
</li>
<li>
<b>HIGH_CARD_SHUFFLE</b> generates a sequence of numeric "cards" for a game.
</li>
<li>
<b>HIGH_CARD_SIMULATION</b> simulates a game of choosing the highest card in a deck.
</li>
<li>
<b>I4_POWER</b> returns the value of I^J.
</li>
<li>
<b>I4_UNIFORM_AB</b> returns a scaled pseudorandom I4 between A and B.
</li>
<li>
<b>I4VEC_MAX</b> returns the value of the maximum element in an I4VEC.
</li>
<li>
<b>PERM_UNIFORM_NEW</b> selects a random permutation of N objects.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 26 February 2014.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>