



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 
