Probability in Games What is Probability Probability is

Probability in Games

What is Probability? Probability is a branch of mathematics that deals with calculating the likelihood an event will occur and is usually expressed as a percentage or number between 1 and 0.

Probability • Game designers try to provide players with a new experience each time a game is played. • Randomness is built into the mechanics of a game, and varies the number and location of obstacles that a player may encounter. • Randomness, probability, likelihood and chance are all terms that describe how variety and unexpected outcomes are created in games.

Non-electronic game probability • Board games use dice, spinners and sandtimers, etc. to add a level of randomness and conflict to games. • When players must carry out a move within a limited amount of time, or when a player’s move is calculated by the roll of dice, the game experience is made unpredictable.

Non-electronic game probability • Game designers use their knowledge of probability to plan the randomness that occurs in their game. • The probability of rolling any number with a pair of dice helps the game designer plan where to put large payoffs or huge danger zones. • In most games, large payouts or huge dangers should occur infrequently.

Types of Probability in Games • • • Dice Spinner Number of objects Size of objects Speed of objects

Dice • The use of dice goes back thousands of years. • Dice were probably originally made from joint bones of animals. • Later dice were made from bone, metal, or pottery. Old Dice make from knucklebones from sheep or goats.

Dice • In ancient times, the outcome of a dice throw was not considered luck, but believed to be controlled by the gods. • For this reason, casting dice was a way of dividing inheritances. • In ancient Rome, the goddess Fortuna, the daughter of Zeus (Lady Luck), was believed to determine the outcome of a throw.

Kinds of Dice

Probability Using 2 Dice • First, assume each die is a cube with 6 equal sides. • Second, assume each die has numbers 1 -6. Opposite numbers on the die always equal 7.

Dice in Games • Dice can come in many shapes and sizes besides the typical 6 sided types. The number of choices changes with the number of sides on a dice. Let’s look at a few of the different types of dice.

Spinner use in Games • When a game designer decides to include a spinner in a game, the design and layout of the spinner will affect the frequency of outcomes. • Spinners created for small children may use pictures, colors, or symbols instead of numbers. • Spinners are frequently chosen to be used in games for young children rather than dice because the small, hard dice pose a choking hazard.

Spinner use in Games • Poor spinner construction and the way a player interacts with a spinner can significantly change the outcome. • The number of choices on the wheel changes the percentage of outcomes. spin and change the outcomes.

Spinner Probability – The size and number of individual choices on the wheel will change the percentage of outcomes. – Thus the probability of an event is the total number of ways an event can occur divided by the total number of outcomes.

Spinner Click image to visit Java Activity

Types of Probability • Empirical Probability: The estimated number of times an event will occur. • Theoretical Probability: The actual number of times an event does occur.

Randomness in card games • Occurs when players “shuffle” the cards, or when the designer chooses how many suits are within the deck. • The more frequently a card is turned up or played affects the outcome of the game. • Players use their knowledge of probability to calculate the likelihood that a certain event will occur to plan their next move.

Randomness in card games • In poker, the value of the various hands increases as the probability of the hand occurring goes down. It is very rare for a player to get a Royal Flush (Ace, King, Queen, Jack, 10 all in the same suit).

Probability of Being Dealt a Poker Hand 5 card deal 5/52 X 4/51 X 3/50 X 2/49 X 1/48 = 1/2, 598, 960 • The first fraction is the chance of getting any one of 5 particular cards in the deck (5 in 52, since there are 52 cards in the poker deck). • The second fraction is your chance of getting any one of 4 particular cards in the deck on your second card, and so on through the fifth fraction or the fifth card. • By multiplying them together you find out what the chances are of coming up with any hand of unique cards. • The numerator of our fraction in this case represents our royal flush in spades, while the denominator represents all the combinations possible.

Probability of Being Dealt a Poker Hand 5 card deal Poker Hand Number of Combinations Probability ----------------------------Royal Straight Flush 4 . 0000015391 Other Straight Flush 36 . 0000138517 Four of a kind 624 . 0002400960 Full House 3, 744 . 0014405762 Flush 5, 108 . 0019654015 Straight 10, 200 . 0039246468 Three of a kind 54, 912 . 0211284514 Two Pairs 123, 552 . 0475390156 One Pair 1, 098, 240 . 4225690276 High card only 1, 302, 540 . 5011773940 Total 2, 598, 960 1. 00000 5/52 X 4/51 X 3/50 X 2/49 X 1/48 = 1/2, 598, 960

Probability of Multiple Choice • There was a TV game called “Lets Make a Deal”. • Contestants had to chose between 3 different doors. • Behind one of the doors was a great prize. • Usually, there was a “clunker” behind one of the doors.

Number of Doors Click image to visit Java Activity

Coin Toss Probability of heads or tails when Flipping 1 coin. Probability of unordered pairs when flipping 2 coins. Probability of ordered pairs when flipping 2 coins.

Coin Flip Probability

Coin Toss Click image to visit Java Activity

Crazy Choice Click image to visit Java Activity

Experimental Probability Click image to visit Java Activity

Probability of Pulling Marbles Out of a Bag Click image to visit Java Activity

Two Colors Click image to visit Java Activity

Probability in Monopoly The probability of landing on a specific property in Monopoly is a factor of the probability of the dice, the location of the property on the board, the randomness of Community Chest and Chance cards, and if players pay to get out of jail.

Name Probability Rank In Jail 3. 9499 1 Marvin Gardens 2. 5861 22 Illinois Avenue 3. 1858 2 Pennsylvania Avenue 2. 5006 23 Go 3. 0961 3 Virginia Avenue 2. 4649 24 New York Avenue 3. 0852 4 Short Line 2. 4326 25 B & O Railroad 3. 0659 5 States Avenue 2. 3721 26 Reading Railroad 2. 9631 6 Community Chest 2. 3661 27 Tennessee Avenue 2. 9356 7 Income Tax 2. 3285 28 Pennsylvania Railroad 2. 92 8 Vermont Avenue 2. 321 29 Free Parking 2. 8836 9 Connecticut Avenue 2. 3003 30 Kentucky Avenue 2. 8358 10 Visiting Jail 2. 2695 31 Water Works 2. 8074 11 Oriental Avenue 2. 2621 32 St. James Place 2. 7924 12 Park Place 2. 1864 33 Indiana Avenue 2. 7357 13 Luxury Tax 2. 1799 34 Atlantic Avenue 2. 7072 14 Baltic Avenue 2. 1624 35 St. Charles Place 2. 7017 15 Mediterranean Avenue 2. 1314 36 Ventnor Avenue 2. 6789 16 Community Chest 1. 8849 37 Pacific Avenue 2. 6774 17 Chance 1. 048 38 Boardwalk 2. 626 18 Chance 0. 8669 39 North Carolina Avenue 2. 6252 19 Chance 0. 865 40 Electric Company 2. 604 20 Go To Jail 0 41 Community Chest 2. 5945 21

Players tend to pay to get out of jail early in the game when there is property to buy. This increases their chances of purchasing property.

Players tend to stay in jail longer after much of the property has been developed to avoid landing on opponents’ property. It’s safe in jail. This changes the probability dynamics of the game.

These numbers represent the amount of money you will earn or pay out every time you roll. For example: There are 40 squares on the Monopoly board and the average amount on a typical roll of the dice is “ 7”. If you divide 40 by 7 you will get about 6. Take the $200 for passing Go and divide by 6 turns and you will get $33. 78 for every roll taken in the game. Taking into account the amount you pay on taxes, you will earn $29. 81 for every turn you take throughout the game.

Conclusions on Monopoly • Railroads are excellent investments, particularly when owned together. • The best return on investment is from putting a third house on New York Avenue. It has the fastest payoff of any building on almost all of the properties. • The square most landed on other than Jail is Illinois Avenue, a hotel there will bring the most income other than a hotel on Boardwalk. • The worst investment is to buy Mediterranean Avenue without first owning Baltic Avenue. • The properties between the Jail square and the Go To Jail square landed on the most, because of the jump caused by landing on Go To Jail. • The Orange ones have the biggest bang for the buck as far as building goes.

Be sure to draw the right conclusion!

Batting Averages Your favorite baseball player is batting 300. What does this mean? A batting average involves calculating the probability of a player's getting a hit. The sample space is the total number of at-bats a player has had, not including walks. A hit is a favorable outcome. Thus if in 10 at-bats a player gets 3 hits, his or her batting average is 3/10 or 30%. For baseball stats we multiply all the percentages by 10, so a 30% probability translates to a 300 batting average.

Batting Averages Career Records Career Leaders & Records for Batting Average

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