Doubled Card Values

Solution & Explanation

To solve the problem of determining the probability that one card's value is exactly double that of another in a deck of cards numbered from 1 to 100, we need to follow these steps:

1. Understanding the Problem:

   • We have a deck of cards numbered 1 through 100.
   • We draw two cards randomly.
   • We want to find the probability that the number on one card is exactly twice the number on the other card.

2. Identifying Possible Pairs:

   • For every card numbered x, the corresponding card that would make it a "doubled pair" is 2x.
   • The maximum value for x such that 2x is still in the deck (1 to 100) is 50 (since 2 * 50 = 100).
   • Therefore, the valid pairs are: (1, 2), (2, 4), (3, 6), ..., (50, 100).
   • This gives us 50 valid pairs.

3. Total Number of Possible Combinations:

   • We need to consider the total number of ways to draw two cards from a deck of 100 cards.

   • Using the combination formula $C(n, k) = \frac{n!}{k!(n-k)!}$ where $n = 100$ and $k = 2$:

     $C(100, 2) = \frac{100!}{2!(100-2)!} = \frac{100 \times 99}{2} = 4950$

   • So, there are 4950 possible pairs of cards.

4. Calculating the Probability:

   • The probability $P$ that one card's value is exactly double the other is the ratio of favorable outcomes (doubled pairs) to the total possible outcomes (all pairs).

     $P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{50}{4950}$

   • Simplifying this fraction gives:

     $P = \frac{1}{99} \approx 0.0101$

5. Conclusion:

   • The probability that one card's number is exactly twice the number on the other card is approximately 0.0101 or 1.01%.

Key Points:

• Combination Concept: The use of combinations ensures that order doesn't matter when drawing two cards.
• Pair Formation: Identifying valid pairs where one number is double the other is crucial.
• Probability Calculation: This involves dividing the number of successful outcomes by the total number of possible outcomes.

By following this structured approach, we can confidently determine the probability of drawing two cards such that one card's value is exactly double the value of the other.