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

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To calculate the probability that the arithmetic mean of four drawn cards equals 405, one must determine the probability that the sum of the four cards equals 1620. This can be approached using the Stars and Bars theorem, which helps in finding the number of ways to achieve this sum. The formula involves calculating the number of combinations that satisfy the equation divided by the total possible outcomes, which is 500 raised to the power of 4. The resulting probability is approximately 0.011. This method provides a structured way to analyze the problem of drawing cards with specific numerical constraints.
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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.
 
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pizzico85 said:
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:
$$\begin{align*}P(\text{arithmetic mean is 405})&=P(X_1+X_2+X_3+X_4=4\cdot 405) \\
&=\frac{\text{Number of ways that }X_1+X_2+X_3+X_4=1620}{500^4} \\
&= \frac 1{500^4}\binom{1620-1}{4-1} \\
&\approx 0.011
\end{align*}$$
 
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