Problem 7 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

An experiment consists of selecting a card at random from a 52-card deck. Refer to this experiment and find the probability of the event. A king of diamonds is drawn.

Step-by-Step Solution

Answer

The probability of drawing a King of Diamonds from a 52-card deck is \(P(\text{King of Diamonds}) = \frac{1}{52}\).

Step 1: Identify the total number of possible outcomes

We have a standard 52-card deck, so there are 52 possible cards that could be drawn.

Step 2: Identify the successful outcomes

Only one card in the deck is a King of Diamonds, so there is only 1 successful outcome.

Step 3: Calculate the probability

Now, divide the number of successful outcomes (1) by the total number of possible outcomes (52): \(P(\text{King of Diamonds}) = \frac{\text{Number of successful outcomes}}{\text{Total number of possible outcomes}} = \frac{1}{52}\)

Step 4: Simplify the probability

The probability of drawing a King of Diamonds from a 52-card deck is: \(P(\text{King of Diamonds}) = \frac{1}{52}\)

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arranging, and finding patterns. It is incredibly useful for calculating probabilities, especially in situations involving numerous possibilities and outcomes. In our exercise, combinatorics is at play when we deal with the deck's 52 possible cards.

• Combinatorics helps us identify successful outcomes; in this case, only one card satisfies our condition—the King of Diamonds.
• When calculating probabilities, we often use combinations and permutations, which are core principles of combinatorics.

Understanding these basics allows us to assess the chance of specific events happening, such as finding the likelihood of drawing particular cards from a deck. We count the number of favorable outcomes against the total possible number, crafting the foundation for probability calculations.

Statistics

Statistics is all about gathering, analyzing, and interpreting data. In probability, statistics provide the tools we need to make sense of random events. Our exercise here is a straightforward demonstration of a probabilistic event analyzed with statistical methods.

• In this scenario, selecting cards is a statistical experiment—it involves making observations and drawing conclusions based on data (the cards drawn).
• The probability calculated demonstrates a simple statistical measure, offering predictions on how often events occur on average under controlled circumstances.

By breaking down this data, statistics helps us understand random phenomena, allowing us to make informed guesses about uncertain situations, like drawing a specific card from a deck.

Card Probability

Card probability is a specialized area of probability focused on events involving playing cards. Since a deck has a fixed number of cards, it's an excellent opportunity to learn probability basics. Let's explore this using the exercise.

• Each card drawn from a standard deck is an independent random event since the outcome of one draw doesn't affect the others.
• The probability of drawing the King of Diamonds, calculated as \(\frac{1}{52}\), illustrates how probabilities of individual cards are determined based on their total count.

Card probability lets us predict outcomes and understand the likelihood of drawing specific cards. This knowledge can be applied in games and real-life decisions, improving strategic thinking and decision-making skills.