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

In summary, the probability of the arithmetic mean of the sum of the numbers on four cards being 405 with four people each drawing from a deck of 500 cards is approximately 0.011. This can be calculated using the Stars and Bars theorem, which states that the number of ways to distribute identical objects into distinct groups can be represented using combinations. For a more detailed explanation and visual representation, please refer to the Stars and Bars theorem.
  • #1
pizzico85
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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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  • #2
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:
$$\begin{align*}P(\text{arithmetic mean is 405})&=P(X_1+X_2+X_3+X_4=4\cdot 405) \\
&=\frac{\text{Number of ways that }X_1+X_2+X_3+X_4=1620}{500^4} \\
&= \frac 1{500^4}\binom{1620-1}{4-1} \\
&\approx 0.011
\end{align*}$$
 

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

What is the formula for calculating the arithmetic mean of the sum of random variables?

The formula for calculating the arithmetic mean of the sum of random variables is:
Mean of sum = Mean of first random variable + Mean of second random variable + ... + Mean of nth random variable

How does the arithmetic mean of the sum of random variables relate to the probability distribution of the variables?

The arithmetic mean of the sum of random variables is equal to the sum of the means of each individual random variable. This means that the mean of the sum is directly related to the probability distribution of the variables, as the mean is a measure of central tendency and is influenced by the shape of the distribution.

Can the arithmetic mean of the sum of random variables be used to predict future outcomes?

The arithmetic mean of the sum of random variables can be used to make predictions about future outcomes, but it is important to note that it is not a guarantee. It is based on the assumption that the variables are independent and identically distributed, which may not always be the case in real-world scenarios.

How does the sample size affect the calculation of probability with arithmetic mean of the sum of random variables?

The larger the sample size, the more accurate the calculation of probability with arithmetic mean of the sum of random variables will be. This is because a larger sample size provides a more representative sample of the population, reducing the likelihood of sampling error and resulting in a more reliable estimate of the mean.

Can the arithmetic mean of the sum of random variables be negative?

Yes, the arithmetic mean of the sum of random variables can be negative. This can occur if the individual random variables have negative values and the sum of these values results in a negative mean. However, it is important to consider the context of the problem to determine if a negative mean makes sense in the given scenario.

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