Problem 2 - 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 spade.

Step-by-Step Solution

Answer

The probability is \( \frac{4}{17} \).

Step 1: Understanding the Problem

In this problem, we are dealing with conditional probability. We need to find the probability that the second card drawn is a spade, given that the first card drawn was a spade.

Step 2: Establish Total Outcomes after First Draw

Initially, the deck contains 52 cards, of which 13 are spades. If we've already drawn a spade as the first card, this leaves us with 51 total cards remaining.

Step 3: Determine Successful Outcomes for Second Draw

After drawing one spade, there are 12 spades remaining in the deck of 51 cards. Thus, the number of favorable outcomes where the second card is a spade is 12.

Step 4: Calculate the Conditional Probability

The probability that the second card is a spade given that the first card is a spade is calculated as the ratio of successful outcomes over the total number of outcomes after one spade has already been drawn. This probability is \( \frac{12}{51} \).

Step 5: Simplify the Probability Fraction

Simplify \( \frac{12}{51} \) by dividing the numerator and the denominator by their greatest common divisor, which is 3. This gives us \( \frac{4}{17} \).

Key Concepts

Probability Theory

Probability theory is a branch of mathematics that deals with the likelihood of different outcomes occurring. At its core, it helps us understand how likely an event is to happen. Probability is often expressed as a fraction or a decimal between 0 and 1, where 0 indicates impossibility, and 1 denotes certainty. For instance, if we say there is a probability of 0.5 that it will rain tomorrow, it means there is a 50% chance of rain.

Conditional probability, a fundamental concept within probability theory, describes the probability of an event occurring given that another event has already happened. It refines the likelihood of an event by incorporating additional information. For example, knowing that a player has drawn a spade from a deck first, shapes our understanding of the likelihood that the next card is also a spade. It tells us, "Given we know something has happened, how does it affect what we are predicting?"

In mathematical terms, conditional probability is usually expressed as \( P(B|A) \), which translates to "the probability of event B happening given that event A has already occurred." This can be computed by the formula: \( P(B|A) = \frac{P(A \cap B)}{P(A)} \). Understanding conditional probability is crucial for solving problems that involve dependencies between events.

Card Deck Probability

Dealing with probabilities involving a deck of cards is a classic way to apply the principles of probability theory. A standard deck contains 52 cards, split evenly into 4 suits—spades, hearts, diamonds, and clubs—each with 13 cards. Calculating probabilities with cards often requires understanding and counting the number of favorable outcomes for a given scenario.

For instance, finding the probability of drawing a spade from a full deck is straightforward: since there are 13 spades out of 52 cards, the probability is \( \frac{13}{52} \), which simplifies to \( \frac{1}{4} \). However, when a card is drawn, the situation changes because the deck is altered. If one spade is already drawn, only 51 cards remain, and just 12 are spades. This needs recalculating probabilities based on the new card count.

Moreover, conditional probability alters our calculations because it considers already drawn cards. Continuing our earlier example—having drawn one spade—the probability the next card is a spade becomes \( \frac{12}{51} \). This formula derives from our redefining the universe of possibilities after the first draw. It's an important approach to mastering card-related probability problems.

Simplifying Fractions

Once a probability is calculated, it's often represented as a fraction. Simplifying fractions is the process of reducing the numerator and the denominator to their smallest equivalent values. This makes the fraction easier to interpret and compare with others.

To simplify a fraction, we find the greatest common divisor (GCD) of the numerator and the denominator. With our example, the probability of drawing a spade on the second draw given the first was a spade is \( \frac{12}{51} \). To simplify, notice both 12 and 51 are divisible by 3. Dividing both by their GCD yields \( \frac{4}{17} \). Hence, \( \frac{4}{17} \) is the simplest form of that probability fraction.

Simplifying fractions is vital because it allows you to present the most reduced and hence easily comparable form of probabilities. It helps in understanding and communicating the likelihood in a tidy and standard way.