Probability in Games

1. Probability in Games

2. 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.

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

4. Kinds of Dice

5. 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.

6. 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.

7. 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.

8. 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.

9. 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

10. Dice Table Click image to visit Java Activity

11. Racing Game With One Die Click image to visit Java Activity

12. Racing Game With Two Dice Click image to visit Java Activity

13. 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.

14. 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.

15. Spinner Click image to visit Java Activity

16. Spinner Again Click image to visit Java Activity

19. 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.

20. Number of Doors Click image to visit Java Activity

22. 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.

23. Coin Flip Probability

24. Coin Toss Click image to visit Java Activity

25. Crazy Choice Click image to visit Java Activity

26. Experimental Probability Click image to visit Java Activity

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

29. Two Colors Click image to visit Java Activity

30. 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.

32. 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.

33. 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.

35. 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.

36. 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.

38. Be sure to draw the right conclusion!

39. Probability of Being Dealt a Poker Hand5 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.

40. Probability of Being Dealt a Poker Hand5 card deal Poker Hand      Number of Combinations   Probability--------------------------------------------------------Royal Straight Flush                 4        .0000015391Other Straight Flush               36        .0000138517Four of a kind                    624        .0002400960Full House                      3,744        .0014405762Flush                           5,108        .0019654015Straight                       10,200        .0039246468Three of a kind               54,912        .0211284514Two Pairs                     123,552        .0475390156One Pair                    1,098,240        .4225690276High card only           1,302,540        .5011773940Total                        2,598,960       1.0000000000 5/52 X 4/51 X 3/50 X 2/49 X 1/48 = 1/2,598,960

41. 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.

42. Batting Averages Career Records Career Leaders & Records for Batting Average