Problem 56 - College Algebra with Modeling and Visualization

Question

Conditional Probability and Dependent Events Find the probability of drawing a king from a standard deck of cards given that two cards, both kings, have already been drawn and not replaced.

Step-by-Step Solution

Answer

The probability is \( \frac{1}{25} \).

Step 1: Understanding the Problem

We have drawn two cards from a deck and both are kings. We need to calculate the probability of drawing a king on the next draw from the remaining deck.

Step 2: Determine Total Outcomes

After drawing and not replacing two kings, there are now 50 cards left in the deck (52 original cards - 2 kings drawn = 50 cards).

Step 3: Determine Favorable Outcomes

Originally, there were 4 kings in the deck. After drawing two, there are 2 kings remaining in the deck.

Step 4: Calculate Conditional Probability

The probability of drawing one of the remaining kings is the number of remaining kings divided by the total number of remaining cards: \( \frac{2}{50} \).

Step 5: Simplify the Fraction

Simplify \( \frac{2}{50} \) to get the final probability. Thus, \( \frac{2}{50} = \frac{1}{25} \).

Key Concepts

Probability Theory

Probability theory is a fundamental branch of mathematics that deals with the likelihood or chance of different outcomes. It provides a framework for understanding random events and can be applied to various fields like statistics, finance, science, and more. In a basic sense, probability measures how likely an event is to occur, expressed as a number between 0 and 1, where 0 means the event will not happen and 1 indicates certainty.

The concept involves a few key components:

• Sample Space: This is the set of all possible outcomes of an experiment. For example, when flipping a coin, the sample space is {Heads, Tails}.
• Event: An event is any subset of the sample space. For instance, rolling a die and getting a number less than 3 is an event.
• Probability of an Event: This is the ratio of the number of favorable outcomes to the total number of possible outcomes. Expressed mathematically as \( P(A) = \frac{n(A)}{n(S)} \), where \( n(A) \) is the number of favorable outcomes and \( n(S) \) is the total number of outcomes in the sample space.

The concept discussed here - conditional probability - is a key concept in probability theory. It examines the probability of an event occurring given that another event has already taken place. This is particularly useful when dealing with dependent events.

Dependent Events

In probability, events are considered dependent when the occurrence of one event affects the likelihood of another. This is the opposite of independent events, where events do not impact each other. Dependent events often occur in scenarios where items or events are not replaced after they occur or are selected.

For instance, consider drawing cards from a deck without replacement. The probability of drawing a specific card changes as you draw more cards since the composition of the deck is altered. If you draw a king and don't replace it, the number of kings left in the deck decreases, influencing the probability of drawing another king.

To calculate probabilities for dependent events, we use conditional probability, which is represented as \( P(A | B) \), meaning the probability of event A occurring given that event B has already occurred. The formula is:
\[P(A | B) = \frac{P(A \cap B)}{P(B)}\]
For the exercise, after drawing two kings, the conditional probability was calculated by acknowledging the changed conditions: two kings were no longer available, and the total deck had fewer cards.

Card Combinatorics

Card combinatorics is a specific application of combinatorial mathematics focused on calculating probabilities and outcomes related to card games and card selections. It uses principles of probability, especially when dealing with a standard deck of 52 cards, to solve problems involving possible combinations and arrangements of cards.

When dealing with card combinatorics, it’s crucial to understand the deck's composition and how it impacts probability calculations. A standard deck has four suits (hearts, diamonds, clubs, and spades) and each suit has 13 ranks, including four kings.

Relevant concepts include:

• Permutations: These are arrangements where the order matters. For example, in card games, dealing cards in a specific order could be vital.
• Combinations: A way of choosing items where the order does not matter. For instance, selecting a hand of five cards is a combination problem.

In the exercise, understanding that two of the four kings were already drawn helps in solving the problem using combinatorial counting principles. Thus, with 50 cards left and only 2 kings remaining, the probability calculation was based on the reduced numbers from these combinatorial considerations.