Problem 1 - Calculus for Biology and Medicine

Question

Suppose you draw 2 cards from a standard deck of 52 cards. Find the probability that the second card is a spade given that the first card is a club.

Step-by-Step Solution

Answer

The probability is \( \frac{13}{51} \).

Step 1: Understand the Problem

We are asked to find the probability that the second card is a spade given that the first card drawn is a club. We begin by understanding the total number of outcomes and the favorable outcomes.

Step 2: Number of Spades in the Deck

There are 13 spades in a standard deck of 52 cards. Since we're given that the first card is a club, this doesn't affect the number of spades left in the deck.

Step 3: Adjust Deck for the First Card Being a Club

Since the first card is a club, it means there are only 51 cards remaining in the deck after removing the first club card from the original 52.

Step 4: Calculate the Probability

The probability that the second card is a spade, given that the first is a club, involves simply considering the remaining spades out of the now 51 total cards. The probability is given by \( \frac{13}{51} \).

Step 5: Conclusion

We find the probability of the second card being a spade, given the first card was a club, is \( \frac{13}{51} \).

Key Concepts

Probability Theory

Probability theory is a branch of mathematics that explores the likelihood of events occurring. It provides tools to analyze situations where the outcome is uncertain, which is often represented as the probability. In our exercise, we are dealing with conditional probability.
Conditional probability is the chance an event will happen given that another related event has already occurred. It's denoted as \( P(A | B) \), where \( A \) is the event of interest and \( B \) is the event that has already happened.
In our card problem, we are interested in calculating the probability that the second card is a spade (event \( A \)), given that the first card was a club (event \( B \)). This sort of calculation is common in probability theory as it helps solve real-world problems involving sequences of events.

Playing Cards

Playing cards are a common tool used in probability exercises. A standard deck consists of 52 cards divided into four suits: spades, hearts, diamonds, and clubs. Each suit has 13 cards. Understanding the structure of a deck is crucial when calculating probabilities related to card games.
In our exercise, we specifically look at the suits—spades and clubs. The problem involves drawing two cards sequentially from the deck and observing which suits they belong to. To solve this, it's important to note:

• Cards are not replaced after being drawn, reducing the pool of remaining cards.
• The initial draw affects the probability of subsequent draws.

These principles of card drawing will differ if one uses more complicated decks or rules, but for our standard deck, these are essential considerations.

Combinatorics

Combinatorics is a field of mathematics dealing with combinations, arrangements, and counting. It's widely used to determine the number of possible outcomes in a situation or the arrangement of objects. In the given exercise, combinatorics helps us count the total and favorable outcomes when drawing cards.
The exercise requires recalibrating the possible outcomes based on an initial condition. Combinatorics tells us that:

• There are originally 52 possible cards to be drawn.
• Once one card is drawn and revealed as a club, only 51 cards remain.
• The count of spade cards stays at 13, as drawing a club won't affect their number.

Hence, combinatorics was implicitly used to adjust the deck size and determine the revised probability of drawing a spade after a club.