Doubling Numbers

Solution & Explanation

To solve this problem, we need to determine the probability that one card's number is exactly double the other card's number when two cards are drawn from a deck numbered from 1 to 100.

Step 1: Understanding the Problem

• Total Cards: 100 cards numbered from 1 to 100.
• Objective: Find the probability that one card's number is double the other.

Step 2: Identifying Favorable Outcomes

• Favorable Pairs: For a card with number X, the only possible card that can double it is 2X. Thus, we have pairs like:
  • (1, 2), (2, 4), (3, 6), ..., (50, 100)
• Number of Favorable Pairs: 50 pairs (since the maximum value for X is 50 to ensure 2X is not greater than 100).

Step 3: Calculating Total Possible Outcomes

• Total Ways to Choose Two Cards: This can be calculated using combinations:

  $\binom{100}{2} = \frac{100 \times 99}{2} = 4950$

Step 4: Calculating Probability

• Probability Formula:

  $P(\text{one card is double the other}) = \frac{\text{Number of favorable outcomes}}{\text{Total possible outcomes}}$

• Substitute Values:

  $P = \frac{50}{4950} = \frac{1}{99}$

Step 5: Verification

• Double Counting Consideration: The problem considers unordered pairs, so each pair (X, 2X) is counted once, and the reverse (2X, X) is not a separate pair since order doesn't matter in combinations.

Conclusion

The probability that one card's number is exactly double the other card's number is $\frac{1}{99}$. This solution uses basic probability principles by identifying favorable outcomes and dividing by the total number of possible outcomes.