Problem 3 - Algebra Form and Function

Question

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

Step-by-Step Solution

Answer

Answer: The probability of picking a red card from a standard deck of playing cards is 50%.

Step 1: Identify the number of red cards in the deck

In a standard deck of 52 playing cards, there are two red suits – hearts and diamonds. Each suit contains 13 cards, thus making a total of 26 red cards. So, the number of red cards = 13 (hearts) + 13 (diamonds) = 26 cards.

Step 2: Calculate the ratio of red cards to the total number of cards in the deck

We need to find the ratio of red cards to the total number of cards. The ratio can be written as the fraction: Red cards / Total number of cards In our case, the ratio would look like this: 26 red cards / 52 total cards

Step 3: Simplify the ratio/fraction if possible

Since both the numerator (red cards) and denominator (total cards) are even numbers, we can simplify the fraction. Dividing both by 2, we get: (26/2) / (52/2) 13 red cards / 26 total cards

Step 4: Convert the ratio to a percentage

To convert the ratio or fraction into a percentage, we multiply the result by 100. (13 red cards / 26 total cards) * 100 0.5 * 100 = 50% The probability of picking a red card from a deck is 50%.

Key Concepts

Playing Cards

Playing cards are a fascinating tool not just for games but also for understanding probability. A standard deck of playing cards consists of 52 individual cards. These cards are divided equally into four suits: hearts, diamonds, clubs, and spades.

• The hearts and diamonds are referred to as the red suits, each consisting of 13 cards.
• The clubs and spades are called black suits, also with 13 cards each.

The richness in variety of cards allows us to explore many probability scenarios simply and effectively. Since each suit has the same number of cards, it helps us maintain an equal distribution for various probability calculations.

Ratios

A ratio is a mathematical expression that compares two numbers or quantities. In probability, ratios are often used to compare the number of favorable outcomes to the total number of possible outcomes. For example, when we want to determine the probability of drawing a red card from a deck, we start by writing this as a ratio:

• Number of red cards (favorable outcomes): 26
• Total number of cards in the deck (possible outcomes): 52

This gives us the ratio \( \frac{26}{52} \), indicating that out of every 52 cards, 26 are red. We can simplify this ratio by finding the greatest common divisor, which is 2 in this case, reducing it to \( \frac{13}{26} \). Ratios provide a foundational step in understanding and conveying probability.

Percentages

Percentages are a common way to represent probability, often making the data easier to understand at a glance. It tells us how many parts of a hundred represent our interest. Converting a probability ratio to a percentage involves a simple calculation. Taking the simplified ratio from above, \( \frac{13}{26} \), the conversion to percentage is done by multiplying the ratio by 100.

• \( \frac{13}{26} \times 100 = 50\% \)

Thus, there is a 50% probability of picking a red card from a deck, meaning out of any 10 times, you can expect to pick a red card 5 times. Percentages translate probabilities into a more intuitive and relatable format, making it easier to envisage likelihood in daily life.