# Probability in Games

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```					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 number
between 1 and 0.
Types of Probability in Games
   Dice
   Spinner
   Number of objects
   Size of objects
   Speed of objects
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.
Probability Using 2 Dice

This table shows all of the possible outcomes of
rolling 2 dice. In the top row, one die is a 1 and the
other squares shows the possible outcomes of the
other die. Notice that it is possible to have squares
with the same values, but in different orders.
Probability Using 2 Dice

Next, find the sum of the two dice in each square. What we are
looking for is the number of times the SUM of both dice occurs.
Probability Using 2 Dice
X is equal to the value in each cell of the previous table. Count is
the number of occurrences (times or likelihood) of that sum. Find
the sum of all the counts – which in this case is 36. In the right
column is the probability of the outcome and it is calculated by
dividing the individual count by the total count. For example, the
likelihood of rolling two dice and getting a total of 7 is 6 times out
of 36 events or 1 in 6.
Dice in RPG Games
• Dungeons & Dragons is credited with popularizing the use of dice in RPG’s.
• RPG’s use dice to determine the outcome of events, such as the success or
failure
• This is considered a fairer decision since success or failure are decided
randomly
• Games typically determine success as either a total on one or more dice
above or below a target number, or a certain number of rolls above a
certain number.
• Dice can also be used by a game master for other purposes, such as to
randomly generate game content or to make arbitrary decisions.
• Some games use dice to determine what attributes the player's character
has when created
Dice Table

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Racing Game With One Die

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Racing Game With Two Dice

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Spinner use in Games
   Spinners are a good choice for small children.
They are easy to understand. Dice are a choking
hazard for small children who love to put things
into their mouth.
   Spinners are not always constructed well.
   Spinners do not often spin “true”.
   Players can change the effort they put into their
spin and change the outcomes.
Spinner Probability

   The number of choices
on the wheel changes the
percentage of outcomes.
   The size of individual
choices on the wheel will
change the percentage of
outcomes.
Spinner

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Spinner Again

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Probability of Multiple Choice
   There was a TV game
called “Lets Make a
Deal”.
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

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Coin Toss Probability
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

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Crazy Choice

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Experimental Probability

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Probability of Pulling Marbles Out of a Bag

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Two Colors

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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
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
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
are 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!
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      1.0000000000

5/52 X 4/51 X 3/50 X 2/49 X 1/48 = 1/2,598,960
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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