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Or
Unions

 

 Now let's talk about using the word OR.

 
 What is the chance of getting a heads
 OR a tails when you flip a coin?
 
 Well hey, 
 the coin is either going to come up heads
 or it is going to come up tails.
 That means there is a 100% chance
 of getting a heads or a tails.
 

 100% = 1

 
 The chance of getting a heads is ...
 

 
 The chance of getting a tails is ...
 

 
 It looks like we have a pattern here too!
 It looks like when we have the word OR
 with independent events, we just add up the fractions.
 Like this ...
 
 The chance of a coin flip coming up heads OR tails is ...
 

 
 That does work ... sometimes.

 But sometimes it doesn't work.

 Watch ...
 
 What is the chance of getting a heads on the first flip of a coin
 OR a heads on the second flip of the coin?
 

 If we use the "just add them up" plan, we get ...

 
 The chance of getting a heads on the first flip
 or the second flip is ...
 

 

 A 100% chance means that he thing will absolutely happen.

 But hey, both flips COULD come up tails.
 So the chance that either the first flip will be heads
 OR the second flip will be heads CAN'T be 100%.
 
 So what's the deal? Why didn't this one work?
 
 It doesn't work because the two things can overlap.

 That is, both flips could be heads.

 We could draw a picture of this ...
 

 
 So now we have to decide just exactly what we mean

 when we say OR.

 Do we mean:
 One thing happens or the other thing happens or both.
 Or do we mean:
 One thing happens or the other thing happens but not both.
 
 Long ago, somebody decided that what OR means is ...
 "One thing happens or the other thing happens OR BOTH."
 
 If they wanted to leave out the "or both" part, they call it an ...
 

 EXCLUSIVE OR

 
 That name is pretty long, 
 so sometimes they shorten it to ...
 

 XOR

 
 Review Time:
 
AND One AND the other.
 
OR One OR the other OR both
 
XOR One or the other BUT NOT BOTH
 
 When we figure out the ones that use OR,
 we need a way not to count the overlap part twice.
 

 
 If you look close, you will see that the overlap part
 is where the first flip comes up heads
 and the second flip comes up heads too.
 
 To get rid of the double counting.
 That means the chance of one thing OR the other
 (or both) happening, is ...
 

 
 So for our coin flips, we have ...
 

 

 
 So there is a 75% chance that the first flip,
 or the second flip, or both, will be a heads.
 
 There is another way to look at this OR calculation
 that might be easier to understand.
 If it is, great. If not, forget it. At least for now.
 Here it is ...
 
 Think about what you don't want to happen.
 You are looking for the chance of a heads 
 on the first flip OR the second flip.
 The only way you don't get that is if you get a tails 
 on the first flip AND the second flip
 Rather than calculating the OR with all it's overlaps to worry about,
 just calculate the other side, THE AND, and subtract that from 100%.
 
 An AND is just a multiplication.
 
 Example:
 
 What's the chance of getting a heads on the first or second coin flip.
 The opposite of this is getting a tails on the first AND second flip, 
 so calculate that and subtract it from 100%.
 
 The chance of getting a tails on the first flip is 50% ...
 The chance of getting a tails on the second flip is 50% ...
 
 The chance of getting a tails on the first AND second flips is ...
 

 50% 50% = 25%

 
 So the chance of this NOT happening, is ...
 

 100% - 25% = 75%

 
 And if we didn't get a tails on the first and second flips,
 that means one or the other or both was a heads.
 The same answer, what a shock!
 

   copyright 2005 Bruce Kirkpatrick

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