Problem 6 - Algebra Form and Function

Question

Give the probability, as a percentage, of picking the indicated card from a deck. Spade

Step-by-Step Solution

Answer

Answer: 25%

Step 1: Determine the number of spades in the deck

In a standard deck of 52 playing cards, there are 4 suits: spades, clubs, hearts, and diamonds. Each suit contains 13 cards (A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K). Therefore, there are 13 spades in the deck.

Step 2: Calculate the probability of drawing a spade

To calculate the probability of drawing a spade, we need to divide the number of spades (13) by the total number of cards in the deck (52). Using the formula for probability, P(spade) = (number of spades) / (total number of cards): P(spade) = \frac{13}{52}

Step 3: Simplify the fraction

We can simplify the fraction by dividing both numerator and denominator by the common divisor 13: P(spade) = \frac{13}{52}=\frac{13/13}{52/13}=\frac{1}{4}

Step 4: Convert the probability to a percentage

To convert the probability to a percentage, we multiply the probability fraction by 100: Probability\,percentage = \frac{1}{4} *100= 25\% The probability of picking a spade from a standard deck of 52 playing cards is 25%.

Key Concepts

Playing Cards and Suit Distribution

A standard deck of playing cards is an essential element within card games and probability problems. Understanding its basic composition can greatly aid in solving related exercises.

• A typical deck contains 52 individual cards.
• These cards are equally divided into four suits: spades, clubs, hearts, and diamonds.
• Each suit holds 13 cards, numbered from Ace through 10 and including three face cards: Jack, Queen, and King.

Knowing that each suit has an equal distribution of 13 cards is fundamental when calculating probabilities, such as the likelihood of drawing a specific card suit like spades from the deck.

Fraction Simplification in Probability

Finding probabilities often leads us to work with fractions. Simplifying these fractions makes it easier to understand and interpret probabilities.

• When you encounter a fraction, check if both the numerator and denominator have a common factor.
• In this exercise, the fraction \(\frac{13}{52}\) represents the probability of picking a spade.
• To simplify, divide both 13 and 52 by their greatest common divisor, which is 13, simplifying the fraction to \(\frac{1}{4}\).

Simplifying fractions removes complexity from the numbers involved and makes performing further calculations, like percentage conversion, more straightforward.

Percentage Conversion of Probabilities

Probabilities are often expressed as percentages to provide a clearer understanding of how likely an event is to occur.

• To convert a simplified fraction to a percentage, multiply the fraction by 100.
• In the example where the simplified probability of drawing a spade is \(\frac{1}{4}\), you multiply by 100: \(\frac{1}{4} \times 100 = 25\%\).
• This conversion communicates the probability of picking a spade more intuitively: there's a 25% chance out of 100% total possibilities.

Converting fractions to percentages helps in comparing probabilities and making quick, informed decisions based on the likelihood of different outcomes.