Problem 19 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the probability for the experiment of selecting one card at random from a standard deck of 52 playing cards. The card is a red face card.

Step-by-Step Solution

Answer

The Probability of selecting a red face card from a deck is \( \frac{3}{26} \).

Step 1: Determine the Total Outcomes

Determine the number of total possible outcomes. In a playing deck of cards, there are 52 cards, so the total outcomes are 52.

Step 2: Find the Successful Outcomes

Next, determine the successful outcomes. In a deck of cards, there are 4 face cards (King, Queen, Jack) in each suit (Hearts, Diamonds, Clubs, Spades). But we're only considering the red face cards. Hearts and Diamonds are the red suits, so for each there are 3 red face cards. Therefore, there are 3+3=6 red face cards in total.

Step 3: Apply Probability Formula

Now, apply the formula for probability, which is Probability = Successful outcomes / Total Outcomes. By substituting our values we get Probability of picking a red face card = 6/52 = 3/26.

Key Concepts

Total Outcomes

When dealing with probability problems, it is important to first establish the *total outcomes*. This refers to the complete set of possible results in an experiment or event. For example, if you are dealing with a standard deck of playing cards, each card represents a unique possibility, leading to a total of 52 possible outcomes. To break it down further, a standard deck of cards includes:

• 4 suits: Hearts, Diamonds, Clubs, and Spades
• Each suit contains 13 cards
• Total of 52 cards (4 suits x 13 cards per suit)

Identifying the total outcomes ensures you have the right base figure needed to calculate probabilities accurately.

Successful Outcomes

In probability, *successful outcomes* are the results we are specifically interested in when conducting an experiment. These are the outcomes that meet the criteria set by the problem. Let's consider our exercise: finding the probability of randomly selecting a red face card from a standard deck of 52 cards. Face cards are defined as Kings, Queens, and Jacks. For red face cards, we only look at hearts and diamonds:

• Each suit has 3 face cards: King, Queen, and Jack
• Hearts (a red suit) has 3 face cards
• Diamonds (another red suit) also has 3 face cards

Adding those, we find 6 red face cards in total (3 from Hearts + 3 from Diamonds). Hence, the successful outcomes are 6. Properly understanding which outcomes are successful lets you effectively use them in probability calculations.

Probability Formula

The core of probability calculations lies in using the *Probability Formula*. This is a simple ratio of successful outcomes to total outcomes, providing an exact measure of the likelihood of a specific event occurring.The formula is expressed as:\[\text{Probability} = \frac{\text{Successful Outcomes}}{\text{Total Outcomes}}\]Applying this to our card-deck problem:

• We identified 6 successful outcomes (red face cards)
• The total outcomes available are 52 cards in the deck

So, the probability of drawing a red face card is:\[\text{Probability} = \frac{6}{52} = \frac{3}{26}\]This simplified ratio tells us that there's a 3 in 26 chance of randomly selecting a red face card from the deck. Understanding how to apply this formula is crucial for tackling a wide range of probability exercises.