



Writing
out a big coin flip tree diagram is no fun.



It
would be great if there was a way to get the job done 


without
all that drawing. 





Good
news! 


Here
it is ... 





To
use the shortcut, 


you
need to know how to use something called a factorial. 





A
factorial is like a little computer program 


that
we run on a number. 


Here's
how it works. 





Say
you want to do a factorial on the number 4. 


What
we do is multiply together all of the numbers from 1 to 4. 


That
is ... 





4
× 3 × 2 × 1






4 × 3 = 12, 12
× 2 = 24, 24 × 1 = 24 





So
... 





4 × 3 × 2 × 1
= 24 





Math
people just love using little codes to save space. 


The
code they use to say: 


"Do
the factorial thing with this number" 


is
the exclamation point "!" 





So
...






4! means 4 × 3
× 2 × 1 





4! = 24 





Example:






What
is 6! equal to? 





6! = 6 × 5 × 4
× 3 × 2 × 1 





6 × 5 =
30, 30 × 4 = 120, 120 × 3 = 360, 


360 × 2 = 720,
720 × 1 = 720






Lot's
of calculators have a key to do factorials. 


Most
of the time, the key looks like this ... 











So
great, 


now
we know what a factorial is.



So
what? 


What
do we do with it? 





We
do this ... 





Example: 





Say
we want to know the chance



of
getting exactly 4 heads in 10 coin flips. 





The
first thing we need to know, 


is
how many combinations of heads and tails 


there
are in 10 flips. 





We've
done that kind of thing before. 


It
went like this ... 











To
do this, 


it
really helps if you have a calculator with this key ... 











Anyway,
however you do it, figure out that 2^{10} = 1024. 


So
there are 1024 paths in 10 coin flips. 





Next
we need to know how many of them have exactly 4 heads. 


Here's
how that works. 


We
finally get to use the factorial deal. 


The
formula is ... 











So
for 4 flips ... 











There
are 210 paths that give us exactly 4 heads. 


There
are 1024 total paths. 


The
chance of getting exactly 4 heads is ... 











So
the chance of getting exactly 4 heads in 10 flips 


is
about 20%. 





Example: 





What
is the chance of getting AT LEAST 8 heads in 10 coin flips? 





At
least 8 heads means 8 or 9 or 10 heads. 


We
need to figure out the chance of each of these 


and
then add them together. 





We've
already figured out that the number of paths 


in
10 coin flips is 





2^{10}
= 1024






The
number of paths with exactly 8 heads are ... 











So
the chance of getting 8 heads is ... 











Now
figure out the number of paths 


with
exactly 9 heads ... 











So
the chance of getting 9 heads in 10 flips is ... 











Now
figure out the number of paths with 10 heads. 











WHOA!
HOLD ON! WAIT A MINUTE!!!






What
does 0! mean? 





It
works like when we have 0 as an exponent. 





3^{0}
= 1 12^{0} = 1 402^{0} =
1 AND ... 0! = 1






So
great. 


0!
= 1. 


That
means we have ... 











If
you think about it, that shouldn't be a surprise. 


How
many ways can all 10 flips come up heads? 





So
the chance of getting all 10 heads is ... 











So
all together, the chance of getting 8 or 9 or 10 heads 


(remember
"OR" means we add) is ... 





8.8%
+ 1.0% + 0.1% = 9.9%






So
there is about a 10% chance of getting 8 or more heads 


in
10 coin flips. 





Pop
Quiz: 


What
is the chance of NOT getting 8 or more heads 


in
10 coin flips? 










































































copyright 2005 Bruce Kirkpatrick 
