



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 
