My brother and I are entered into a lottery of ten people total
(including ourselves).  Names will be pulled out of a hat, the 1st
name through 7th name are the losers.  The 8th name pulled is 3rd
place winning $20, the 9th name pulled is 2nd place winning $30 and
the 10th name pulled is 1st place winning $50.  What is the
probability that either my brother or me will win 1st, 2nd or 3rd
place?

I know that if the 1st, 2nd and 3rd names pulled were winners the
formula for figuring out the problem would be:

1 - 8/10*7/9*6/8

What is the probability and formula backing the answer for figuring
out the original way the question was stated?

I know the chance of heads is 1/2 the first time I flip a coin, but
what is it the last time?

you have one draw, there are 10 options available, 3 winning options,
7 losing. 3/10 chance you win. also 3/10 your brother wins. combined,
you have a 6/10 chance of either of you winning. do your own homework!

Yes, do your own homework, but don't listen to marcus, or you'll get a bad grade.

This isn't homework, haha.  I'm a 26 year old defense contractor,
school is a distant memory.  I'm just trying to educate myself on a
real world situation.

Ok, well the probability you list in your question is correct for any
three randomly drawn names.  First three, last three, any three. 
Essentially, the non-winning picks don't 'count'.  You don't even have
to look at them.

The probability that either you or you brother or both will win one of
the three prizes is:

1 - [Probability you both lose]

As Rracecarr mentions, it doesn't matter where in the ten names drawn
the three winners lie, the probability is the same.  So your formula

1 - 8/10*7/9*6/8

is correct.

No one seems to have quite the correct answer. It is 1 minus the
probability of both you and your brother losing. Losing is the same as
the chance of picking 2 numbers less than or equal to 7 from a pile of
numbers 1 thru 10. This is 7 choose 2 divided by 10 choose 2, that is,
C(7,2)/C(10,2) which equals 7*6/(10*9). So the answer is 1 -
7*6/(10*9).

uhhh....

That's the same answer as handlit's, ansel's, and mine.