Calculation of probability with arithmetic mean of the sum of random variables

[pizzico85]

Calculation of probability with arithmetic mean of random variables

There are 4 people, each of whom has one deck of cards with 500 cards that are numbered from 1 to 500 with no duplicates.

Each person draws a card from his deck and I would like to calculate the probability of the event that "the arithmetic mean of the number on the 4 cards is 405".

How to make that?Some explanation is welcome.

 

[I like Serena]

There are 4 people, each of whom has one deck of cards with 500 cards that are numbered from 1 to 500 with no duplicates.

Each person draws a card from his deck and I would like to calculate the probability of the event that "the arithmetic mean of the sum of the number on the 4 cards is 405".

How to make that?Some explanation is welcome.

Hi pizzico85, welcome to MHB! ;)

It's an application of the Stars and Bars theorem.
It's explained in detail here: Stars and Bars theorem
They explain it better - and with pictures - than I can.

More specifically, you have:
$$\begin{align*}P(\text{arithmetic mean is 405})&=P(X_1+X_2+X_3+X_4=4\cdot 405) \\
&=\frac{\text{Number of ways that }X_1+X_2+X_3+X_4=1620}{500^4} \\
&= \frac 1{500^4}\binom{1620-1}{4-1} \\
&\approx 0.011
\end{align*}$$