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A Shortcut
Using Factorials

 

 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 210 = 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
 

 210 = 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.
 

 30 = 1   120 = 1   4020 = 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

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