Probability of Drawing Exactly Two Cards with the Same Number from a Deck

In a standard deck, the cards are distributed based on one of four colors (red, green, blue, yellow) and numbered from 1 to 7 for each color. This gives us a total of 28 cards. A random draw of 5 cards is taken from this deck. What is the probability that exactly two of these cards share the same number?

Understanding the Deck and Drawing Mechanism

Let's first establish how many total ways we can draw 5 cards from the deck of 28 cards. This follows the combination formula:

[ C_{28}^5 frac{28!}{5! times; 23!} 98280 ]

Calculating the Favorable Outcomes

The goal is to have exactly two cards out of the five share the same number. We can calculate the number of ways to achieve this as follows:

Step 1: Choosing the Number

There are 7 different numbers, and we need to choose one. This can be done in:

[ C_7^1 7 ]

Step 2: Choosing the Colors for the Two Cards

The chosen number can appear in any of the 4 colors. We need to select 2 out of the 4 colors, which can be done in:

[ C_4^2 6 ]

Step 3: Choosing the Numbers for the Remaining Three Cards

Now, we need to select 3 different numbers from the remaining 6. This can be done in:

[ C_6^3 20 ]

Step 4: Assigning Colors to the Remaining Three Numbers

Each of these 3 chosen numbers can appear in any of the 4 colors, resulting in:

[ 4^3 64 ]

Thus, the total number of ways to achieve this is:

[ 7 times; 6 times; 20 times; 64 53760 ]

Calculating the Probability

The probability is given by the ratio of the number of favorable outcomes to the total number of outcomes:

[ P frac{53760}{98280} ≈ 0.5470 ]

This can also be expressed as approximately 54.70%.

To summarize, the probability that exactly two cards out of five randomly drawn will share the same number is calculated as:

[ frac{53760}{98280} ≈ 0.5470 ]

Conclusion

This problem demonstrates the application of combinatorial principles in calculating the probability of specific outcomes in a finite sample space. Understanding these principles is crucial for various fields, including statistics, data science, and even real-world applications in games and lotteries.