Problem 11 - 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 red face card is drawn.An ace is not drawn.

Step-by-Step Solution

Answer

The probability of drawing a red face card is \(\frac{3}{26}\) and the probability of not drawing an ace is \(\frac{12}{13}\).

Step 1: Understanding a 52-card deck

A standard 52-card deck contains four suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards: - 9 number cards from 2 to 10 - 3 face cards (jack, queen, and king) - 1 ace Hearts and diamonds are red, while clubs and spades are black.

Step 2: Event 1: A red face card is drawn

A red face card can either be a face card from the hearts suit or the diamonds suit. There are 3 face cards in each suit (jack, queen, and king). Therefore, there are 3 + 3 = 6 red face cards in the deck. The total number of possible outcomes is 52, as there are 52 cards in the deck. Therefore, the probability of drawing a red face card is: \(P(\text{red face card}) = \frac{\text{number of red face cards}}{\text{total number of cards}} = \frac{6}{52} = \frac{3}{26}\)

Step 3: Event 2: An ace is not drawn

In a 52-card deck, there are 4 aces (one from each suit). To find the probability of not drawing an ace, we first calculate the number of outcomes where an ace is not drawn: Total non-ace cards = Total number of cards - Number of aces = 52 - 4 = 48 So, the probability of not drawing an ace is: \(P(\text{not an ace}) = \frac{\text{number of non-ace cards}}{\text{total number of cards}} = \frac{48}{52} = \frac{12}{13}\) In conclusion, the probability of drawing a red face card is \(\frac{3}{26}\) and the probability of not drawing an ace is \(\frac{12}{13}\).

Key Concepts

52-card Deck Composition

Grasping the structure of a 52-card deck is the foundation from which all card-based probability questions are answered. A standard deck contains 52 cards, which are divided evenly into four suits: hearts, diamonds, clubs, and spades. Each suit is represented by a different symbol and color; hearts and diamonds are red, while clubs and spades are black.

Every suit in the deck has the same composition: 13 cards consisting of nine numbered cards (2 through 10), three face cards (jack, queen, king), and one ace. This grouping is crucial for calculating the odds of drawing any particular type of card. For instance, knowing there are only three face cards per suit can help you determine the likelihood of drawing a face card, while the singularity of aces in each suit plays into the calculation of avoiding an ace.

Calculating Probability in Card Experiments

The probability of an event is a measure of how likely it is to occur, expressed as a ratio between the number of favorable outcomes and the total number of possible outcomes. In terms of a card experiment, each card represents a possible outcome.

To compute probability, use the formula:
\( P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}} \).

When calculating the likelihood of drawing a particular card from a deck, the denominator is always 52, since there are 52 unique cards. The numerator varies depending on the specific card or type of card you're interested in. In the case of drawing a red face card, the favorable outcomes are those specifically red face cards, which we've identified as a countable few. Similarly, determining the probability of not drawing an ace involves counting every card that isn't an ace.

Card Suit and Values

The details of card suit and values are elemental to understanding card game dynamics and, by extension, to solving probability problems involving a deck of cards. Each of the four suits—hearts, diamonds, clubs, and spades—holds an equal share of the deck, with 13 cards each. However, the value hierarchy within each suit is where some might get confused.

Face cards, labelled jack, queen, and king, typically hold a higher value than numbered cards, which range from 2 to 10. The ace is unique; in many games, it can either be the highest card, surpassing the king, or the lowest, acting as a 1. Understanding these values influences the way we calculate probabilities. Knowing, for example, that there are more numbered cards than face cards could affect the way we approach a problem or the strategy we use in a game.