



We
can draw a diagram of a coin flip.



It
looks like this ... 











You
read the picture from left to right. 


From
the "start" place on the left, we can take either path. 


Unless
something tells us otherwise, 


the
chance of taking either path is equal. 


One
path goes to heads, meaning the coin flip came up heads. 


The
other path goes to tails, meaning the coin flip came up tails. 





What
happens if we flip the coin more than one time? 





If
we flip the coin two times, it could come up ... 


Heads
and then heads (2 heads). 


Heads
and then tails (1 head and 1 tail). 


Tails
and then heads (1 tail and 1 head). 


Tails
and then tails (2 tails). 





If
we care about the order that the flips happened, 


these
are 4 different results. 


If
we only care about the total of heads and tails, 


there
are only three different results. 


The
middle two would be the same result. 


If
you wanted to be really picky, you could say they were equivalent
results. 





If
we wanted to draw a picture of the two flips, 


it
would look like this ... 











If
we have a fair coin, 


the
chance of getting either a head or a tail on any flip are the same. 


We
have the same chance of taking any path on this tree. 


That
means we have the same chance of arriving at the end of any on these
paths. 











So
the chance of getting any one of these results is 25 percent. 





Most
of the time, we don't care what order the flips happen. 


That
means we can add the two middle results together, 


and
say the chance of getting one head and one tail is 50 percent. 





What
would happen if we flipped the coin three times? 


How many
ways could it come out? 


Let's see
... 











Eight Ways! 





That
means there is a 1 in 8 chance (also known as 12.5% chance) 


of
any particular one of these paths being followed. 





If
we don't care what order we get the results, 


and
most of the time we don't, 


we
can combine the ones that have the same totals ...












There
are a couple of things to make sure that you see here ... 





1)
The chance of getting three heads, 


is
the same as the chance of getting three tails 





2)
The chance of getting two heads and one tails 


is
the same as the chance of getting two tails and one heads 





3)
There are eight different paths 


from
the start to the end of three flips. 


That's
because there are two ways that each flip can come out 


and
there are three flips. 


That
is ... 











We
only had a couple of coin flips here, 


so
it wasn't too hard to draw the chart 


of
how the flips would come out. 





But
if we had 20 flips, the chart would be very big. 


There
would be ... 





2^{20}
= 1,048,576






A
chart with over one million paths would be a pain to draw. 





copyright 2005 Bruce Kirkpatrick 
