Introduction

Understanding the intricacies of probability is crucial for many fields, including statistics, gambling, and game design. One common problem in probability involves drawing cards from a deck while considering the impact of not replacing the drawn card. Let's explore the probability of drawing a 3 on the first draw and then a heart on the second draw without replacement.

Step-by-Step Solution

When drawing cards from a standard deck of 52 cards, the problem requires us to calculate the combined probability of two specific events happening in sequence without replacement. The events are: Drawing a 3 on the first draw. Drawing a heart on the second draw.

Probability of the First Event

The probability of drawing a 3 from a full deck of 52 cards is:

P(Drawing a 3) 4/52 1/13

This is because there are 4 threes in a deck of 52 cards.

Probability of the Second Event

After drawing a 3, we do not replace it, so there are now 51 cards left in the deck. Among these, 13 are hearts. However, if the first card was a 3 of hearts, there will be 12 hearts left. Let's consider both scenarios:

P(Drawing a heart | First card is not hearts) 13/51

P(Drawing a heart | First card is a heart) 12/51

Combining the Probabilities

The overall probability is the weighted sum of these two scenarios:

1. Probability of drawing a 3 of any suit except hearts and then a heart:

P(3 of non-hearts and heart) (4/52) * (13/51) (1/13) * (13/51) 1/51

2. Probability of drawing the 3 of hearts and then a heart:

P(3 of hearts and heart) (1/52) * (12/51) (1/52) * (12/51) 12/(52 * 51) 1/221

Adding these probabilities together:

P(3 first and heart second) (1/51) (1/221) (221 51) / (51 * 221) 272 / (51 * 221) 1/52

Alternative Approach

We can also approach this problem from a different perspective, one that avoids explicit computations. Imagine we select our preferred card by drawing two cards in sequence without replacement. We define our favorite card as having the value of the first card and the suit of the second card. Given that the deck is well-shuffled and every card is equally likely to be the first card, and that the second card thus determines the suit, every card is equally likely to be selected as our favorite card. Therefore, the probability that our favorite card is the ace of hearts is 1/52, as every card has an equal chance of being selected.

Conclusion

Whether we use the traditional method of calculating probabilities or the more intuitive approach of considering the equal likelihood of each card being chosen, the result remains the same. The probability of drawing a 3 on the first draw and a heart on the second draw, without replacement, is 1/52. This problem demonstrates the elegance and consistency of probability principles, providing valuable insights into how probabilities can be calculated and interpreted in different contexts.