Problem 14 - Basic College Mathematics

Question

Playing Cards. In Exercises \(13-18\), assume that one card is randomly selected from a well-shuffled deck (see p. 460). Find the probability that the card is a picture card (jack, queen, or king).

Step-by-Step Solution

Answer

\( \frac{3}{13} \)

Step 1 - Identify the total number of cards in a deck

A standard deck has 52 cards in total. This includes 4 suits: hearts, diamonds, clubs, and spades, each with 13 cards.

Step 2 - Count the number of picture cards

There are 3 picture cards (jack, queen, and king) in each suit. Since there are 4 suits, the total number of picture cards is: \[3 \text{ cards/suit} \times 4 \text{ suits} = 12 \text{ picture cards}\]

Step 3 - Calculate the probability

The probability of drawing a picture card from a well-shuffled deck can be calculated by dividing the number of picture cards by the total number of cards in the deck: \[\text{Probability} = \frac{12 \text{ picture cards}}{52 \text{ total cards}} \]

Step 4 - Simplify the fraction

Simplify the fraction: \[\frac{12}{52} = \frac{3}{13}\]. So, the probability is: \[\frac{3}{13}\]

Key Concepts

probability calculation

Probability represents the likelihood of an event happening. It's expressed as a fraction or a percentage. The basic formula for calculating probability is:

\(\text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}\).

To apply this concept to the problem of drawing a picture card from a deck, we need two key pieces of information:

• The total number of cards in the deck (52).
• The number of picture cards (12).

Once we have these numbers, we divide the number of picture cards by the total number of cards. This fraction tells us how likely it is to draw a picture card from the deck.

standard deck

A standard deck of playing cards consists of 52 cards. These 52 cards are divided into four suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards, ranging from numbers 2 to 10, and including an Ace, King, Queen, and Jack.

• Hearts: A red suit.
• Diamonds: Another red suit.
• Clubs: A black suit.
• Spades: Another black suit.

Understanding the structure of a standard deck is essential for any probability calculation involving playing cards. In our specific problem, recognizing that there are exactly 3 picture cards (Jack, Queen, King) in each suit helps to simplify our calculations.

picture cards

Picture cards, also known as face cards, include the Jack, Queen, and King. In each of the four suits of a standard deck, there is one of each picture card.

• Each suit (hearts, diamonds, clubs, spades) contains one Jack, one Queen, and one King.
• Since there are 4 suits, we have 3 picture cards per suit and a total of 12 picture cards in the deck.

To find the probability of drawing a picture card from a deck, we need to know their total compared to the total number of cards. In this case, it's 12 picture cards out of 52 cards in total.

Simplifying this fraction \(\frac{12}{52}\) gives us \(\frac{3}{13}\). This means that if you were to shuffle the deck well and draw one card at random, there is a \(\frac{3}{13}\) chance that it will be a picture card.