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The Six Sided Coin
Non Binary Tree Diagrams

 

 A fair coin has an equal chance of coming up heads or tails.

 It has two possible paths on each flip.
 A die (half a pair of dice) has six possible paths.
 You can roll a 1, 2, 3, 4, 5, or 6.
 We can draw a tree diagram of this.
 We get ...
 

 
 If we roll the die two times, 
 how many paths would there be?
 
 We could draw a picture, 
 but that would be kind of messy.
 We could also use our formula ...
 

 
 So there are 36 different paths we can have
 for 2 rolls of 1 die.
 
 How many of those paths add up to 5?
 To add up to 5, we could have ...
 

 4 + 1 = 5

 

 3 + 2 = 5

 

 2 + 3 = 5

 

 1 + 4 = 5

 

 So 4 of the 36 paths add up to 5.
 
 What is the chance of getting a total of 5
 when you roll a die two times?
 

 
 The chance of the number you rolled 
 adding up to 5 is 11.1%
 
 A roll of a die is just like a coin flip in one special way.
 Each roll of the die is it's own game.

 It doesn't matter to a roll of a die,

 what happened on any other roll.
 Each roll is an independent event.
 
 That means it doesn't matter 
 if we roll one die and then roll it again,
 or take two identical dice and roll them at the same time.
 

 OK, back to the problems ...

 
 Example:
 
 What is the chance of rolling a 7 on two dice at a casino?
 

 The ways that you can get a total of 7 on two dice are ...

 
1 + 6 = 7 2 + 5 = 7 3 + 4 = 7
   
4 + 3 = 7 5 + 2 = 7 6 + 1 = 7
 

 There are 6 ways (6 paths) to get a total of 7.

 We said before that we have 36 possible paths

 when we roll two dice.

 That means the chance of getting a 7 is ...

 

 

 There is a 16.7% chance of rolling a 7 on a pair of dice.

 

 Example:
 
 What is the chance of rolling a total of 13 on 2 dice?
 
 The highest number on a die is 6.
 That means the highest total that you can get on 2 dice
 is 6 + 6 = 12.
 So the chance of rolling a 13 is zero.
 Lots of people say ANYTHING IS POSSIBLE.
 But rolling a 13 on 2 dice is NOT.
 
 Tougher Example:
 
 What is the chance of rolling a 4 on three dice?
 
 OK, first figure out all of the paths that add up to 4 ...
 

 1 + 1 + 2 = 4

 

1 + 2 + 1 = 4

 

2 + 1 + 1 = 4

 
 A total of 3 paths add up to 4.
 
 Now figure out how many paths there are all together ...
 

 
 So the chance of rolling a 4 on 3 dice is ...
 

 
 Not too likely, eh?
 

   copyright 2005 Bruce Kirkpatrick

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