Probability of Drawing a Spade from a Modified Card Pack

In a fascinating exploration of probability, let's delve into the scenario where a card is drawn from a standard deck of 52 cards and then placed into a second deck. The question at hand is, what is the likelihood that a card drawn from the second deck is a spade?

The Basic Setup

We start with a standard deck of 52 cards, containing exactly 13 spades, 13 hearts, 13 diamonds, and 13 clubs. When a card is drawn from the first deck and placed into the second deck, the first deck now consists of 51 cards, and the second deck has 52 cards (the one added and the originally present 51 cards).

What we're interested in is the probability P(spade) that a card drawn from the second deck is a spade. The calculation hinges on whether the card transferred from the first deck to the second deck is a spade or not.

Scenario 1: The Transferred Card is a Spade

Let's first consider the scenario where the card transferred to the second deck is a spade. The second deck now has 52 cards, including 14 spades (the 13 original spades plus the one added spade).

The probability that the card drawn from the second deck is a spade in this scenario, denoted as P(A), is:

[text{P(A)} frac{13}{52} times frac{14}{53} frac{13}{52} times frac{14}{53} frac{7}{106}]

Scenario 2: The Transferred Card is Not a Spade

Now, let's consider the scenario where the card transferred to the second deck is not a spade. The second deck will now have 52 cards, 13 of which are spades (the original 13 spades).

The probability that the card drawn from the second deck is a spade in this scenario, denoted as P(B), is:

[text{P(B)} frac{39}{52} times frac{13}{53} frac{39}{52} times frac{13}{53} frac{39}{212}]

Total Probability Calculation

Since these two scenarios (the transferred card being a spade or not a spade) are mutually exclusive, we can add the probabilities calculated for these two scenarios to find the total probability of drawing a spade from the second deck.

[text{P(total)} text{P(A)} text{P(B)} frac{7}{106} frac{39}{212} frac{14}{212} frac{39}{212} frac{53}{212} frac{1}{4} 0.25]

Conclusion

The probability that a card drawn from the second deck is a spade is 0.25 or 25%. This result is interesting because it highlights how the transfer of a single card can impact the probability of drawing a specific card from a modified deck.

The problem can be generalized to other scenarios by applying the principles of conditional probability and the laws of probability. Understanding these concepts is crucial in fields ranging from statistics and data analysis to game theory and machine learning.

For further exploration, students and professionals can experiment with different scenarios and card counts, adjusting the probabilities accordingly. Each variation provides additional insights into the fascinating world of probability theory.

Keywords: Probability of Drawing a Spade, Modified Card Pack, Conditional Probability.