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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 MATLAB program which
simulates a game in which you have one chance to select the highest card
from a deck.
</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">
Usage:
</h3>
<p>
<blockquote>
<b>high_card_simulation</b> ( <i>deck_size</i>, <i>skip_num</i>, <i>trial_num</i> )
</blockquote>
where
<ul>
<li>
<i>n</i> is the number of times the dice are thrown.
</li>
</ul>
</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 = "../../m_src/brownian_motion_simulation/brownian_motion_simulation.html">
BROWNIAN_MOTION_SIMULATION</a>,
a MATLAB program which
simulates Brownian motion in an M-dimensional region.
</p>
<p>
<a href = "../../m_src/dice_simulation/dice_simulation.html">
DICE_SIMULATION</a>,
a MATLAB program which
simulates N tosses of M dice, making a histogram of the results.
</p>
<p>
<a href = "../../m_src/duel_simulation/duel_simulation.html">
DUEL_SIMULATION</a>,
a MATLAB program which
simulates N repetitions of a duel between two players, each of
whom has a known firing accuracy.
</p>
<p>
<a href = "../../m_src/fair_dice_simulation/fair_dice_simulation.html">
FAIR_DICE_SIMULATION</a>,
a MATLAB program which
simulates N tosses of 2 dice, making a histogram of the results.
</p>
<p>
<a href = "../../c_src/forest_fire_simulation/forest_fire_simulation.html">
FOREST_FIRE_SIMULATION</a>,
a C program which
simulates the occurrence of fires and regrowth in a forest,
displaying the results using X Windows,
by Michael Creutz.
</p>
<p>
<a href = "../../m_src/gamblers_ruin_simulation/gamblers_ruin_simulation.html">
GAMBLERS_RUIN_SIMULATION</a>,
a MATLAB program which
simulates the game of gambler's ruin.
</p>
<p>
<a href = "../../m_src/high_card_parfor/high_card_parfor.html">
HIGH_CARD_PARFOR</a>,
a MATLAB program which
uses the parfor statement to compute in parallel the statistics for
a card game in which you are required to guess the location of the
highest card.
</p>
<p>
<a href = "../../m_src/ising_2d_simulation/ising_2d_simulation.html">
ISING_2D_SIMULATION</a>,
a MATLAB 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 = "../../c_src/life_opengl/life_opengl.html">
LIFE_OPENGL</a>,
a C program which
uses OpenGL to display the evolution of John Conway's "Game of Life",
by Simon Green.
</p>
<p>
<a href = "../../m_src/lorenz_simulation/lorenz_simulation.html">
LORENZ_SIMULATION</a>,
a MATLAB program which
solves the Lorenz equations and displays the solution, for various
starting conditions.
</p>
<p>
<a href = "../../f_src/md1/md1.html">
MD1</a>,
a FORTRAN90 program which
carries out a simple molecular dynamics simulation;
</p>
<p>
<a href = "../../f_src/md2/md2.html">
MD2</a>,
a FORTRAN90 program which
carries out a simple molecular dynamics simulation;
</p>
<p>
<a href = "../../f_src/md3/md3.html">
MD3</a>,
a FORTRAN90 program which
carries out a simple molecular dynamics simulation;
</p>
<p>
<a href = "../../f_src/md3glue/md3glue.html">
MD3GLUE</a>,
a FORTRAN90 program which
carries out a simple molecular dynamics simulation;
</p>
<p>
<a href = "../../m_src/poisson_simulation/poisson_simulation.html">
POISSON_SIMULATION</a>,
a MATLAB library which
simulates a Poisson process in which events randomly occur with an
average waiting time of Lambda.
</p>
<p>
<a href = "../../m_src/random_walk_1d_simulation/random_walk_1d_simulation.html">
RANDOM_WALK_1D_SIMULATION</a>,
a MATLAB program which
simulates a random walk in a 1-dimensional region.
</p>
<p>
<a href = "../../m_src/random_walk_2d_avoid_simulation/random_walk_2d_avoid_simulation.html">
RANDOM_WALK_2D_AVOID_SIMULATION</a>,
a MATLAB program which
simulates a self-avoiding random walk in a 2-dimensional region.
</p>
<p>
<a href = "../../m_src/random_walk_2d_simulation/random_walk_2d_simulation.html">
RANDOM_WALK_2D_SIMULATION</a>,
a MATLAB program which
simulates a random walk in a 2-dimensional region.
</p>
<p>
<a href = "../../m_src/random_walk_3d_simulation/random_walk_3d_simulation.html">
RANDOM_WALK_3D_SIMULATION</a>,
a MATLAB program which
simulates a random walk in a 3-dimensional region.
</p>
<p>
<a href = "../../m_src/reactor_simulation/reactor_simulation.html">
REACTOR_SIMULATION</a>,
a MATLAB 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 = "../../m_src/sir_simulation/sir_simulation.html">
SIR_SIMULATION</a>,
a MATLAB program which
simulates the spread of a disease through a hospital room of M by N beds,
using the SIR (Susceptible/Infected/Recovered) model.
</p>
<p>
<a href = "../../m_src/snakes_and_ladders/snakes_and_ladders.html">
SNAKES_AND_LADDERS</a>,
MATLAB programs which
simulate the game of Snakes and Ladders.
</p>
<p>
<a href = "../../m_src/string_simulation/string_simulation.html">
STRING_SIMULATION</a>,
a MATLAB 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 = "../../m_src/three_body_simulation/three_body_simulation.html">
THREE_BODY_SIMULATION</a>,
a MATLAB 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>
<p>
<a href = "../../m_src/traffic_simulation/traffic_simulation.html">
TRAFFIC_SIMULATION</a>,
a MATLAB program which
simulates the cars waiting to get through a traffic light.
</p>
<p>
<a href = "../../m_src/truel_simulation/truel_simulation.html">
TRUEL_SIMULATION</a>,
a MATLAB program which
simulates N repetitions of a duel between three players, each of
whom has a known firing accuracy.
</p>
<p>
<a href = "../../c_src/xising/xising.html">
XISING</a>,
a C program which
models the variations in ferromagnetism in a material, displaying
the results using X Windows.
</p>
<p>
<a href = "../../c_src/xwaves/xwaves.html">
XWAVES</a>,
a C program which
simulates the behavior of solutions of certain forms of the wave equation, displaying
the results using X Windows.
</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_probability.m">high_card_probability.m</a>,
computes the exact probability that a high card strategy that skips
the first K-1 cards and then takes the next card that is higher
will actually select the highest card in the whole deck of N cards.
</li>
<li>
<a href = "high_card_simulation.m">high_card_simulation.m</a>,
the program.
</li>
<li>
<a href = "high_card_shuffle.m">high_card_shuffle.m</a>,
a secondary program, which tries to compute, for small N, a sequence
of integers that could be used as the values of the cards in the
high card guessing game, in a way that guarantees that the values
of the cards seen so far give no clues as to what the maximum card
actually is.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>,
prints the YMDHMS date as a timestamp.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "high_card_simulation_test.m">high_card_simulation_test.m</a>,
calls all the tests.
</li>
<li>
<a href = "high_card_simulation_test01.m">high_card_simulation_test01.m</a>,
estimates the chance of winning with a 100 card deck, and skips of 1, 11,
21, ..., 91 cards.
</li>
<li>
<a href = "high_card_simulation_test02.m">high_card_simulation_test02.m</a>,
computes the chance of winning with a 100 card deck and any skip from 0 to 99,
then makes a plot.
</li>
<li>
<a href = "high_card_simulation_test02.png">high_card_simulation_test02.png</a>,
a plot illustrating the estimated probability of picking the high card
after first skipping 0, 1, 2, ..., 99 cards.
</li>
<li>
<a href = "high_card_simulation_test03.m">high_card_simulation_test03.m</a>,
computes the exact chances of winning with various skips, and a deck of 100 cards.
</li>
<li>
<a href = "high_card_simulation_test03.png">high_card_simulation_test03.png</a>,
plots the exact chances of winning with various skips, and a deck of 100 cards.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 25 May 2012.
</i>
<!-- John Burkardt -->
</body>
</html>