Problem 12 - Algebra and Trigonometry

Question

\(11-12\) . A card is drawn randomly from a standard 52-card deck. Find the probability of the given event. (a) The card drawn is a heart. (b) The card drawn is either a heart or a spade. (c) The card drawn is a heart, a diamond, or a spade.

Step-by-Step Solution

Answer

(a) 0.25, (b) 0.50, (c) 0.75

Step 1: Understanding the Problem

We need to find probabilities involving drawing cards from a standard deck. A standard deck consists of 52 cards, with 4 suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards.

Step 2: Number of Hearts in the Deck

There are 13 hearts in a standard deck.

Step 3: Number of Hearts and Spades

There are 13 hearts and 13 spades, totaling 13 + 13 = 26 cards that are either hearts or spades.

Step 4: Number of Hearts, Diamonds, or Spades

There are 13 hearts, 13 diamonds, and 13 spades, totaling 13 + 13 + 13 = 39 cards that are hearts, diamonds, or spades.

Step 5: Probability of Drawing a Heart (a)

Probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For (a), the probability of drawing a heart is 13 out of 52: \[ P(\text{Heart}) = \frac{13}{52} = \frac{1}{4} \sim 0.25 \]

Step 6: Probability of Drawing a Heart or a Spade (b)

For (b), the probability of drawing a heart or a spade is 26 out of 52:\[ P(\text{Heart or Spade}) = \frac{26}{52} = \frac{1}{2} \sim 0.50 \]

Step 7: Probability of Drawing a Heart, Diamond, or Spade (c)

For (c), the probability of drawing a heart, diamond, or spade is 39 out of 52:\[ P(\text{Heart, Diamond, or Spade}) = \frac{39}{52} = \frac{3}{4} \sim 0.75 \]

Key Concepts

Card Probability

Probability is a mathematical concept that measures the chance or likelihood of a particular event occurring. When it comes to card probability, we often look at scenarios involving drawing cards from a standard deck and determining the chances of certain outcomes. In a deck of 52 cards, the probability can be calculated by dividing the number of successful outcomes by the total number of possible outcomes.

For example, if we are interested in finding the chance of drawing a heart from a deck, we need to determine how many hearts are available. Since each of the four suits (hearts, diamonds, clubs, spades) contains 13 cards, the number of hearts in the deck is 13. Therefore, the probability of drawing a heart can be calculated as:

• Number of successful outcomes: 13 hearts
• Total possible outcomes: 52 cards
• Probability: \( P(\text{Heart}) = \frac{13}{52} = \frac{1}{4} \approx 0.25 \)

This simple ratio helps us determine our chances and is key to understanding the probabilistic nature of drawing cards.

Deck of Cards

A standard deck of cards is a familiar tool used in a wide range of games and probability exercises. Understanding the composition of a deck is crucial for calculating probabilities accurately. A deck consists of 52 cards, divided evenly into 4 suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards that are numbered or face cards including an Ace, numbers from 2 to 10, and the face cards (Jack, Queen, King).

Because a deck is divided into these categories, probability questions often reference the number of cards within each suit or type. For instance, if you were to calculate the probability of drawing either a heart or a spade, you would consider:

• Each of the suits (hearts and spades) has 13 cards
• Total cards in these two suits: \( 13 + 13 = 26 \)
• Total cards in the deck: 52
• Probability: \( P(\text{Heart or Spade}) = \frac{26}{52} = \frac{1}{2} \approx 0.50 \)

By understanding the deck's composition, players can quickly calculate the likelihood of drawing specific cards from the deck.

Suit Probability

Suit probability refers to the chance of drawing a card from a specific suit or combination of suits. In a 52-card deck, since each suit has the same number of cards, calculating the probability for one suit or multiple suits involves counting the relevant cards and dividing by the total number of cards. For instance, consider the scenario where you want to find the probability of drawing a card that is either a heart, diamond, or spade.

To compute this probability accurately:

• Determine the number of cards in each relevant suit: 13 hearts, 13 diamonds, 13 spades
• Total cards in these suits: \( 13 + 13 + 13 = 39 \)
• Total cards in the deck: 52
• Probability: \( P(\text{Heart, Diamond, or Spade}) = \frac{39}{52} = \frac{3}{4} \approx 0.75 \)

Knowing how to calculate suit probability helps in predicting outcomes and making strategy-based decisions in card games or experiments involving card probability.