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Some Folks Go This Way
Decision Trees

 

 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 ...
 

 220 = 1,048,576

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

   copyright 2005 Bruce Kirkpatrick

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