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

#### pizzico85

##### New member
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.

Last edited:

#### Klaas van Aarsen

##### MHB Seeker
Staff member
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*}