Understanding Probability Distribution and Mean in a Card Drawing Experiment

In this article, we will delve into the probability distribution and the calculation of the mean for a specific card drawing experiment. We will use a deck of 20 cards, with each card numbered from 1 to 5, represented four times each. This article will cover the steps to derive the probability distribution and the mean, followed by a summary of the findings.

Introduction to the Experiment

Consider a deck of 20 cards, where each card is numbered from 1 to 5 with each number appearing exactly four times. A random variable, W, is defined as the number on the card drawn from the deck. Our goal is to understand the probability distribution of W and calculate its mean.

Probability Distribution

Given that there are 4 cards for each of the numbers 1 to 5, the probability of drawing a card numbered 1, 2, 3, 4, or 5 is equal. Let's derive the probability distribution of W.

The probability of drawing any specific number (say 1) from the deck is:

$$P(W w) frac{4}{20} frac{1}{5}$$

Since this holds true for each number from 1 to 5, the probability distribution is:

Value of W Probability PW 1 (frac{1}{5}) 2 (frac{1}{5}) 3 (frac{1}{5}) 4 (frac{1}{5}) 5 (frac{1}{5})

Mean of the Distribution

Next, let's find the mean of the distribution of W. The expected value, EW, can be calculated using the formula:

$$E(W) sum_{i1}^{5} i cdot P(W i)$$

Substituting the values:

$$E(W) 1 cdot frac{1}{5} 2 cdot frac{1}{5} 3 cdot frac{1}{5} 4 cdot frac{1}{5} 5 cdot frac{1}{5}$$

$$E(W) frac{1}{5} (1 2 3 4 5) frac{1}{5} cdot 15 3$$

This tells us that the expected value of the number on a randomly drawn card is 3. Hence, if you were to draw a card at random from this deck, the expected value of the number on the card would be 3.

Alternative Calculation of the Mean

Let's re-examine the calculation of the mean of the distribution for clarity and to highlight any potential discrepancies or alternative methods:

The mean can also be calculated using the summation as follows:

$$E(W) 1 cdot frac{4}{20} 2 cdot frac{4}{20} 3 cdot frac{4}{20} 4 cdot frac{4}{20} 5 cdot frac{4}{20}$$

$$E(W) frac{4}{20} (1 2 3 4 5) frac{4}{20} cdot 15 frac{1}{5} cdot 15 3$$

Summary

Probability Distribution: The probability of drawing any specific number (1, 2, 3, 4, or 5) is (frac{1}{5}). Mean: The mean of the distribution of the random variable W is 3.

This indicates that, on average, if you were to draw a card from this deck, the expected value of the number on the card would be 3.

Additional Experiment

Consider the same deck of 20 cards, where each card is numbered from 1 to 5, represented four times each. Now, let's find the mean considering the presence of each numbered card. The mean is defined as:

$$mean frac{1}{20} (4 4 4 4 4) frac{20}{20} 1$$

This calculation is straightforward since each number appears the same number of times, simplifying the mean calculation.

Conclusion

This article has provided a detailed analysis of the probability distribution and the mean of a card drawing experiment using a deck of 20 cards. We have demonstrated that the probability of drawing any specific number is equally (frac{1}{5}), and the mean of the distribution is 3.