Ascending Card Draw

Solution & Explanation

To solve the problem of determining the probability that three cards drawn sequentially from a shuffled deck of 500 cards are in ascending order, we need to consider the following:

Understanding the Problem

• Deck Composition: The deck contains 500 cards, numbered from 1 to 500.
• Task: Draw three cards one after another without replacement.
• Goal: Find the probability that the drawn cards are in strictly increasing order.

Step-by-Step Solution

1. Identify Total Possible Outcomes:

   • When drawing three cards from 500, the order in which the cards are drawn matters initially. Therefore, we calculate the number of ways to draw three cards in sequence:

     $\text{Total ways to draw 3 cards} = 500 \times 499 \times 498$

2. Identify Favorable Outcomes:

   • For any set of three distinct cards, there is exactly one way to arrange them in ascending order. For instance, if the cards drawn are 120, 250, and 475, the only ascending sequence is (120, 250, 475).

   • The number of ways to choose 3 cards from 500 without regard to order is given by the combination:

     $\text{Number of combinations} = \binom{500}{3} = \frac{500 \cdot 499 \cdot 498}{3 \cdot 2 \cdot 1}$

3. Calculate the Probability:

   • The probability that the cards are in ascending order is the ratio of favorable outcomes to total possible outcomes:

     $\text{Probability} = \frac{\binom{500}{3}}{500 \times 499 \times 498}$

   • Simplifying the expression:

     $\text{Probability} = \frac{1}{3!} = \frac{1}{6}$

Explanation

• Why is the Probability 1/6?
  • For any three distinct numbers, there are 3! (factorial of 3) permutations of arranging them. Only one of these permutations is in ascending order.
  • Hence, the probability of drawing three cards in ascending order is always $1/6$ regardless of the size of the deck.

Conclusion

• The size of the deck (500 cards) does not affect the probability of the cards being in ascending order, as the probability is determined by the permutations of any three distinct numbers.
• This is a classic permutation problem where the order of drawing matters but is normalized by the combination when considering the ascending sequence.

This approach clearly outlines the reasoning and calculations necessary to understand and solve the problem, providing a comprehensive explanation for interview preparation.