Probability of Drawing Exactly One Heart: A Simple Card Game

In this article, we will explore a classic card game scenario where Devin and John each draw a card from a standard deck of 52 cards. We will calculate the probability that exactly one of the cards drawn is a heart, using basic principles of probability. This problem illustrates the importance of understanding conditional probabilities and how to combine them to find the total probability.

Scenario and Setup

Devin and John are playing a simple card game involving a standard deck of 52 cards. The deck contains 13 hearts, 13 diamonds, 13 clubs, and 13 spades. Devin draws a card, records its suit, and then replaces it. The deck remains unchanged before John's turn. John then draws a card. We need to determine the probability that exactly one of the cards drawn is a heart.

Breaking Down the Problem

The problem can be broken down into two scenarios:

Devin draws a heart, and John does not draw a heart. Devin does not draw a heart, and John draws a heart.

Scenario 1: Devin Draws a Heart, and John Does Not

Let's start with the first scenario. The probability that Devin draws a heart is:

P(Devin draws a heart) 13/52 1/4

After Devin replaces the card, the deck remains unchanged. Now, the probability that John draws a card that is not a heart is:

P(John does not draw a heart) 39/52 3/4

The combined probability for this scenario is the product of the individual probabilities:

P(Scenario 1) P(Devin draws a heart) × P(John does not draw a heart) (1/4) × (3/4) 3/16

Scenario 2: Devin Draws a Card Other Than Heart, and John Draws a Heart

Next, let's consider the second scenario. The probability that Devin draws a card that is not a heart is:

P(Devin does not draw a heart) 39/52 3/4

Since Devin has replaced the card, the deck is again unchanged. The probability that John draws a heart is:

P(John draws a heart) 13/52 1/4

The combined probability for this scenario is:

P(Scenario 2) P(Devin does not draw a heart) × P(John draws a heart) (3/4) × (1/4) 3/16

Total Probability

To find the total probability that exactly one of the cards drawn is a heart, we add the probabilities of the two scenarios:

P(Exactly one heart) P(Scenario 1) P(Scenario 2) 3/16 3/16 6/16 3/8

Conclusion

The probability that exactly one of the cards drawn is a heart is 3/8. This result is derived by breaking down the problem into simpler components and applying basic principles of probability.

Understanding such scenarios is crucial in the context of probability and can be applied to various real-world situations involving decision-making and risk assessment.