# HIGH_CARD_SIMULATION Guessing What the Highest Card Is

HIGH_CARD_SIMULATION 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.

You are given a deck of DECK_SIZE cards.

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!)

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.

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.

The question is, can you do better than that?

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.

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.

### Languages:

HIGH_CARD_SIMULATION is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

BROWNIAN_MOTION_SIMULATION, a C++ program which simulates Brownian motion in an M-dimensional region.

DUEL_SIMULATION, a C++ program which simulates N repetitions of a duel between two players, each of whom has a known firing accuracy.

FAIR_DICE_SIMULATION, a C++ program which simulates N tosses of 2 dice, making a histogram of the results.

GNUPLOT, C++ programs which illustrate how a program can write data and command files so that gnuplot can create plots of the program results.

ISING_2D_SIMULATION, 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.

POISSON_SIMULATION, a C++ library which simulates a Poisson process in which events randomly occur with an average waiting time of Lambda.

REACTOR_SIMULATION, 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."

STRING_SIMULATION, 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.

THREE_BODY_SIMULATION, 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.

### Reference:

1. Paul Nahin,
Digital Dice: Computational Solutions to Practical Probability Problems,
Princeton University Press, 2008,
ISBN13: 978-0-691-12698-2,
LC: QA273.25.N34.

### List of Routines:

• HIGH_CARD_PROBABILITY: winning probabilities for the high card game.
• HIGH_CARD_SHUFFLE generates a sequence of numeric "cards" for a game.
• HIGH_CARD_SIMULATION simulates a game of choosing the highest card in a deck.
• I4_POWER returns the value of I^J.
• I4_UNIFORM_AB returns a scaled pseudorandom I4 between A and B.
• I4VEC_MAX returns the value of the maximum element in an I4VEC.
• PERM_UNIFORM_NEW selects a random permutation of N objects.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.

Last revised on 26 February 2014.