Problem 5 - Algebra and Trigonometry

Question

A card is drawn randomly from a standard 52-card deck. Find the probability of the given event. (a) The card drawn is a king. (b) The card drawn is a face card. (c) The card drawn is not a face card.

Step-by-Step Solution

Answer

(a) \( \frac{1}{13} \); (b) \( \frac{3}{13} \); (c) \( \frac{10}{13} \).

Step 1: Understand the Total Number of Possible Outcomes

A standard deck contains 52 cards. Therefore, the total number of possible outcomes when drawing one card is 52.

Step 2: Calculate the Probability of Drawing a King

There are 4 kings in a standard deck, one for each suit (hearts, diamonds, clubs, spades). The probability is calculated as the ratio of favorable outcomes to total outcomes: \( \frac{4}{52} = \frac{1}{13} \).

Step 3: Calculate the Probability of Drawing a Face Card

Face cards include jacks, queens, and kings. There are 3 face cards in each suit and 4 suits, so there are a total of 12 face cards. The probability is \( \frac{12}{52} = \frac{3}{13} \).

Step 4: Calculate the Probability of Drawing a Non-Face Card

Since there are 12 face cards, there are 52 - 12 = 40 non-face cards. The probability of drawing a non-face card is \( \frac{40}{52} = \frac{10}{13} \).

Key Concepts

Standard Deck of Cards

A standard deck of cards is pivotal in understanding probability problems related to cards. It consists of 52 cards in total and is divided into four suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards, ranging from ace through to king. This uniform structure makes it convenient for calculating probabilities. For any card drawn from the deck, the total number of possible outcomes is always 52, since that's the fixed number of cards available in a standard deck. Understanding these basics is crucial as they form the foundation for more complex probability calculations.

It's important to remember the classification within each suit:

• Numbers: 2 through 10
• Face cards: Jack, Queen, King
• Ace, which is unique

This recruitment ensures that when you're calculating probabilities, each card type or suit has an equal chance of being drawn, as long as the deck is shuffled well.

Favorable Outcomes

In probability, "favorable outcomes" refer to the specific case or cases that you are interested in. When looking at card probabilities, these outcomes correspond to the cards you want to draw. Understanding what qualifies as a favorable outcome is key to solving probability exercises correctly.

Let's break down a few examples:

• If you're looking for the probability of drawing a king, your favorable outcomes are the 4 kings in the deck.
• Similarly, for face cards, the favorable outcomes are the 12 face cards (3 face cards per suit across 4 suits).

To find the probability, you would divide the number of favorable outcomes by the total number of possible outcomes, which is 52 for a standard deck. This simple division translates your understanding of the card composition into a meaningful probability value.

Non-Face Card Probability

Non-face card probability focuses on all the cards that are not among the jacks, queens, and kings. Since a standard deck has 12 face cards, subtracting these from the total gives us 40 non-face cards. These are the cards that are numbered from 2 to 10, plus the aces, across all four suits.

To calculate the probability of drawing a non-face card, we consider:

• Total non-face cards: 40
• Total cards in the deck: 52

The probability is then derived by dividing the number of non-face cards by the total number of cards in the deck: \[\frac{40}{52} = \frac{10}{13}.\]
This probability calculation highlights how understanding both what constitutes a face card and the structure of a standard deck can help you solve such problems easily.