Problem 22 - Precalculus with Limits

Question

In Exercises 21 - 24, find the probability for the experiment of selecting one card from a standard deck of \( 52 \) playing cards. The card is not a face card.

Step-by-Step Solution

Answer

The probability that the card drawn from a standard deck is not a face card is \( \frac{10}{13} \)

Step 1: Identifying the number of face cards

Identify the number of face cards (Jacks, Queens, Kings) in a standard deck. Each type of face card (Jack, Queen, King) contains 4 cards (one for each suit: hearts, diamonds, clubs, spades). Total face cards = 4 (Jacks) + 4 (Queens) + 4 (Kings) = 12.

Step 2: Calculating the number of non-face cards

As a standard deck contains 52 cards, subtract the total number of face cards from the total cards to get the total number of non-face cards in the deck. Total non-face cards = 52 (total cards) - 12 (total face cards) = 40.

Step 3: Probabilistic calculation

Finally, divide the number of non-face cards by the total cards in the deck to find the probability of drawing a non-face card. Probability = Number of favorable outcomes / Total number of outcomes= 40 / 52.

Step 4: Simplifying the fraction

Now simplify the fraction to the smallest form. Both 40 and 52 can be divided evenly by 4. Hence, the simplified form is 10 / 13.

Key Concepts

Standard Deck of Playing Cards

Understanding the composition of a standard deck of playing cards is essential when engaging in card-based probability exercises. A standard deck contains 52 cards which are divided into 4 suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards, which are further divided into numbered cards from 2 through 10, and face cards which include the Jack, Queen, and King. Additionally, each suit has an Ace, which is sometimes considered a face card due to its unique status, but for the purposes of probability calculations, they are often treated separately. Knowing this breakdown is crucial before calculating the probabilities of drawing specific cards from a deck.

It's important to note that each card in a deck is unique when drawn randomly, meaning no two cards share both their number and suit. This property is what makes calculating probabilities possible, as each draw from the deck without replacement is an independent event with a calculable likelihood based on the remaining composition of the deck.

Calculating Probabilities

Calculating probabilities is a fundamental skill in statistics that involves determining the likelihood of a particular event occurring. The probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. The result is a fraction or a percentage that represents the chances an event will happen. Probabilities always range between 0 and 1 (or 0% to 100%), where 0 means an event is impossible, and 1 denotes certainty.

To ensure accuracy when calculating these ratios, it's imperative to consider all possible outcomes and only count each distinct event once. Clarifying the conditions of the experiment, such as whether replacement is involved or whether certain outcomes are excluded, will impact the total number of possibilities and the probability calculation. Moreover, simplifying the fraction to its lowest terms often aids in interpretation and comparison of probabilities.

Non-Face Cards in Probability

When discussing non-face cards in probability scenarios, we are referring to all the cards in a deck that aren't Jacks, Queens, or Kings. In a standard deck of 52 cards, there are typically 40 non-face cards if we exclude the Ace as a face card. This category includes the numbered cards 2 through 10 in each of the four suits. Calculating the probability of drawing a non-face card involves identifying the total number of these cards (40) and dividing it by the total number of cards in the deck (52).

Understanding the distinction between face and non-face cards helps define the favorable outcomes when determining probabilities in card experiments. The face cards are often the focus of such probability questions because they make up a distinctive subset of the deck, thereby affecting the calculation of chances of drawing other types of cards. The mastery of calculating the probabilities involving non-face cards represents an important foundation in the broader understanding of probability theory.