Problem 4 - Calculus for Biology and Medicine

Question

Suppose you draw 3 cards from a standard deck of 52 cards. Find the probability that the third card is a club given that the first two cards are clubs.

Step-by-Step Solution

Answer

The probability is \( \frac{11}{50} \).

Step 1: Understand the Problem

We need to find the probability that the third card drawn from a deck of 52 cards is a club, on the condition that the first two cards drawn were clubs.

Step 2: Analyze the Deck after Two Draws

Since the first two cards drawn are clubs, there are 11 clubs remaining (13 original clubs minus 2). There are also 50 total cards remaining in the deck (52 original cards minus the 2 drawn).

Step 3: Calculate the Probability of Third Card Being a Club

The probability of the third card being a club is the number of remaining clubs divided by the total number of remaining cards. Therefore, the probability is given by \( \frac{11}{50} \).

Key Concepts

Probability Theory

Probability theory is a branch of mathematics focusing on the analysis of random phenomena. It provides the framework for quantifying the likelihood of various outcomes, making it essential in industries ranging from finance to science. At its core, probability measures how likely an event is to occur, ranging from 0 (impossible) to 1 (certain).
In this exercise, we're dealing with conditional probability, a crucial part of probability theory. Conditional probability considers the likelihood of an event occurring given that another event has already occurred. It's represented mathematically by the formula:

• \( P(A|B) = \frac{P(A \cap B)}{P(B)} \)

Where \( P(A|B) \) is the probability of event A occurring given B is true. However, in our example, since the first two cards are already determined, we directly calculated the probability of the third card being a club without needing to revisit this formula.

Card Drawing

Card drawing problems are a favorite in probability theory because they provide a tangible way to explore probability principles. When you draw cards from a deck, each draw reduces the total number of cards in the deck. This makes card drawing an example of 'without replacement' scenarios, where the probabilities change as each card is drawn.
In our original exercise, we started with a standard deck of 52 cards. Since the first two cards drawn were clubs, only 50 cards remain, and thus only 11 of these are clubs. This shifting context is why the probability for the third card changes dynamically with each card drawn.
Understanding the mechanics of drawing 'without replacement' is vital. It means once a card is drawn, it can't be put back, altering the probabilities for the rest of the deck.

Combinatorics

Combinatorics is the field of mathematics dealing with counting, arrangement, and combination of sets. While the current exercise does not directly require combinatorics calculations, understanding its concepts enriches your grasp of probability. It helps to visualize why certain outcomes have more or less likelihood based on available combinations.
When you draw cards, for example, every unique combination of cards in your hand is a result of combinatorial probability. Although we simply computed a direct probability, combinatorial thinking ensures we understand why there were initially 52 combinations for the first draw but only 50 after drawing two cards.
Basic combinatorial knowledge, such as factorials or the counting principle, builds a solid foundation, preparing you for more complex probability problems involving numerous outcomes to compare.