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Lose the Lottery Big Time
Inverse Probably Calculations

 

 So the chance of choosing all five winning lottery numbers,

 when there are 36 total numbers is about 0.00024%.
  That is, less than 1 in 400,000.
 
 Picking a number that isn't a winner is a lot easier
  than picking a number that is a winner.
 What about picking all 5 numbers,
 and having NONE of them be one of the 5 winning numbers?
 
 Here's how it works.
 There are 36 total numbers; you choose 5 of them.
 That means there are 31 numbers that you didn't choose.
  The chance of one of those being the winning number is ...
 

 
 So there is about an 86% chance that the  first winning number
 will NOT be one of yours.
 OK, what about the next one?
 There are still 35 numbers in the box.
 5 of them are ones you chose,
 so 30 of them are ones you didn't choose.
 The chance that the next winning number 
 is also one that you didn't choose is ...
 

 
 and the chance that the first AND the second winning numbers 
  are not on your ticket is ...
 

 

 You can probably see where this is going.
 The chance of the next three numbers
 also not being on you ticket, given that all the numbers before it
 are also not on your ticket ...

 

 

 
 So the chance that none of the winning numbers
 were on your ticket is ...
 

 

 So there is about a 44% chance that your ticket 

 will have none of the 5 winning numbers on it.
 
 That means there is about a 56% chance (100% - 44%)
 that your ticket will have one or more winning number on it.
 
 Since your ticket is really no different than any other ticket,
 we can generalize this and say that about 44% of all possible tickets

 will have no winning numbers on them.

 We can then also say that about 56% of all possible tickets will have
 one or more winning number on it.
 
 Example:
 

 What is the chance of rolling at least one six

 in 6 rolls of a die?
 The easiest way to figure this out is to find out what the chance is 
 of not rolling any sixes and then ...
 

 100% - that chance = the answer

 
 What is the chance of not getting a 6 on the first roll?
 That means getting a 1,2,3,4,or 5.
 There are 6 numbers on a die,
 and there are 5 of these that we want.
 The chance of rolling one of these is ...
 

 
 This problem is easier than the lottery problem,
 because rolls of a die are independent events.
 A die always has 6 numbers, 
 and any one of 5 of these is what we want.
 The chance of any roll 
 coming up the way we want is always ...
 

 
 So the chance of not getting any 6's 
 on 6 rolls of a die is ...
 

 
 So the chance of not rolling a 6 on any of six rolls is 33.3%
 That means the chance of rolling at least one 6 is ...
 

 100% - 33.3% = 66.7%

 
 So it is twice as likely that you will roll at least one 6
 in six rolls as the chance that you won't roll any 6's.
 

   copyright 2005 Bruce Kirkpatrick

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