Problem 44 - College Algebra

Question

For the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following: A tail on the coin or red ace

Step-by-Step Solution

Answer

The probability is \( \frac{27}{52} \).

Step 1: Identify the Sample Space

When a coin is tossed, there are two possible outcomes: heads (H) and tails (T). A standard deck of cards contains 52 cards, consisting of 4 suits with 13 cards each. Therefore, the total sample space when both a coin is tossed and a card is pulled is the product of the outcomes from both experiments: Sample space size = 2 (coin outcomes) × 52 (card outcomes) = 104 outcomes.

Step 2: Define the Events

We are interested in the event where either a tail is obtained from the coin or a red ace is drawn from the deck. - The event of getting a tail (T) from the coin includes 52 outcomes (one for each card). - The event of drawing a red ace involves 2 possible outcomes (Ace of Hearts and Ace of Diamonds), multiplied by 2 coin outcomes (head or tail), which gives 4 outcomes in total. - Since drawing a red ace and getting a tail can happen simultaneously (2 outcomes), these should not be double counted.

Step 3: Calculate the Probability of the Events

First, calculate the probability of each individual event:- Probability of "a tail on the coin": \[ P(T) = \frac{52}{104} = \frac{1}{2} \]- Probability of "a red ace": \[ P( ext{Red Ace}) = \frac{4}{104} = \frac{1}{26} \]- Probability of "a red ace and a tail": \[ P( ext{Red Ace and T}) = \frac{2}{104} = \frac{1}{52} \]

Step 4: Use the Addition Rule for Probabilities

The probability of either event happening (a tail or a red ace) is found using the addition rule for probabilities, accounting for the overlap:\[ P(T \text{ or Red Ace}) = P(T) + P( ext{Red Ace}) - P( ext{Red Ace and T}) \]Substitute the values calculated:\[ P(T \text{ or Red Ace}) = \frac{1}{2} + \frac{1}{26} - \frac{1}{52} \]

Step 5: Simplify the Expression

Simplify the expression calculated in Step 4:- Convert all probabilities to a common denominator (52 is a common choice): - \( \frac{1}{2} = \frac{26}{52} \) - \( \frac{1}{26} = \frac{2}{52} \)- Substitute them back into the equation:\[ P(T \text{ or Red Ace}) = \frac{26}{52} + \frac{2}{52} - \frac{1}{52} = \frac{27}{52} \]

Step 6: Conclusion: Probability of Event

Therefore, the probability of getting either a tail on the coin or a red ace is \( \frac{27}{52} \).

Key Concepts

Sample Space

The concept of "sample space" is fundamental to understanding probability. It refers to the set of all possible outcomes of an experiment or activity. Let's break it down with the given example of tossing a coin and drawing a card from a deck.
Tossing a coin has straightforward outcomes:

• Heads (H)
• Tails (T)

A standard deck of cards, on the other hand, consists of 52 cards divided into four suits: Hearts, Diamonds, Clubs, and Spades, each having 13 cards.
When you combine these two activities (tossing a coin and drawing a card), you create a combined sample space. The size of this sample space is calculated by multiplying the number of outcomes for each activity.
For this exercise:

• 2 outcomes from the coin (H or T)
• 52 different cards
• Resulting in a total of 104 possible combinations of outcomes.

This combined sample space provides a foundation for calculating probabilities by allowing us to see all possible scenarios.

Addition Rule for Probabilities

The "Addition Rule for Probabilities" is a critical tool for calculating the probability of either of multiple events occurring. It helps to adjust for any overlap between events so that they are not double-counted.
For the exercise, we are interested in finding the probability of either getting a tail on the coin or drawing a red ace from the deck.
Here's how the addition rule steps in:

• The probability of getting a tail (T) when a coin is tossed is calculated as: \[ P(T) = \frac{52}{104} = \frac{1}{2} \]
• The probability of drawing a red ace (Ace of Hearts or Ace of Diamonds) is: \[ P(\text{Red Ace}) = \frac{4}{104} = \frac{1}{26} \]
• Some scenarios include both a red ace and a tail. This situation creates a probability of: \[ P(\text{Red Ace and T}) = \frac{2}{104} = \frac{1}{52} \]

Applying the addition rule:\[ P(T \text{ or Red Ace}) = P(T) + P(\text{Red Ace}) - P(\text{Red Ace and T}) \]This adjustment ensures that combinations of events aren't erroneously counted more than once.

Card Deck

Understanding the structure of a card deck is crucial to solving probability problems that involve playing cards. A standard deck consists of 52 cards, made up of four suits, each containing 13 cards.
These suits include:

• Hearts (red)
• Diamonds (red)
• Clubs (black)
• Spades (black)

Each suit holds an Ace, numbers 2 through 10, and three face cards: Jack, Queen, and King.
Being familiar with this structure allows for accurate probability calculations. In this exercise, focus was on red aces—the Ace of Hearts and Ace of Diamonds. Since both are red, they form a specific set of cards within the deck.
This understanding helps you manage how many possible outcomes relate to drawing a card, which ties into accurately finding and calculating probabilities in combination with other activities, like tossing a coin.