Please help with a sum of dice problem

From: Charlie (gogogo_1001_at_yahoo.com)
Date: 07/08/04 

Date: 7 Jul 2004 18:15:06 -0700

Dear All,

I'm a computer science graduate student, though I've learnt some
calculus and
probability, I'm never good at it. Now, it turns our that I have prove
a equation in
my work, which is very important to me (sum of workloads on parallel
processors), but
to make it more intutitive, and easier to explain, I translated it
into the following
tradition dice problem:

Suppose we have 4 dices (4 processors), but the six faces have
different probability:
(each face represents a workload, i.e. n workloads running on p
processors ...)

1 10%
2 15%
3 25%
4 20%
5 10%
6 20%

let's say, T1,T2,,,,T6 each has probability P1, P2,,,,P6

So the average sum of the 4 dices would be the sum of following:

T1 + T1 + T1 + T1 multiplied by P1 x P1 x P1 xP1
T1 + T1 + T1 + T2 multiplied by P1 x P1 x P1 xP2
T1 + T1 + T1 + T3 multiplied by P1 x P1 x P1 xP3
..... multiplied by ....
T6 + T6 + T6 + T1 multiplied by P6 x P6 x P6 xP1
T6 + T6 + T6 + T2 multiplied by P6 x P6 x P6 xP2
T6 + T6 + T6 + T3 multiplied by P6 x P6 x P6 xP3
T6 + T6 + T6 + T4 multiplied by P6 x P6 x P6 xP4
T6 + T6 + T6 + T5 multiplied by P6 x P6 x P6 xP5
T6 + T6 + T6 + T6 multiplied by P6 x P6 x P6 xP6

However, the sum of above is simply equal to
   4 x (T1 x P1 + T2 x P2 +,,,+ T6 x P6 )

as each dice is identical,

But mathimatically, I couldn't prove this?

Would somebody mind posting a brief answer and, if possible, a brief
explanation of
how it is proved?

Thank you, and I'm sorry if I've wasted your browsing time with this
question.

Charlie Brown,
University of Edinburgh, UK

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