Problem 43 - 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 head on the coin or a club

Step-by-Step Solution

Answer

The probability is approximately 0.625.

Step 1: Identify the Total Outcomes

A coin toss has 2 possible outcomes: heads or tails. A standard deck has 52 cards. Thus, the total number of outcomes for the experiment of tossing a coin and pulling a card is \(2 \times 52 = 104\).

Step 2: Calculate the Probability of a Head

The probability of getting a head when tossing a coin is \(\frac{1}{2}\), since there are 2 possible outcomes and only 1 is a head.

Step 3: Calculate the Probability of Pulling a Club

There are 13 clubs in a standard deck of 52 cards, so the probability of pulling a club is \(\frac{13}{52} = \frac{1}{4}\).

Step 4: Calculate the Probability of a Head and Club

To find the combined probability of both a head and pulling a club, multiply their individual probabilities: \(\frac{1}{2} \times \frac{13}{52} = \frac{13}{104}\).

Step 5: Apply the Addition Rule for Probability

The probability of getting either a head on the coin or a club from the deck is found by adding their probabilities and subtracting any overlap. Use the formula: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). This yields \(\frac{1}{2} + \frac{1}{4} - \frac{13}{104}\).

Step 6: Calculate the Final Probability

Convert probabilities to have a common denominator to combine them: \(\frac{1}{2} = \frac{52}{104}\) and \(\frac{1}{4} = \frac{26}{104}\). Subtract \(\frac{13}{104}\) from the sum of these fractions: \(\frac{52}{104} + \frac{26}{104} - \frac{13}{104} = \frac{65}{104}\). Simplify this to \(\frac{65}{104} \approx 0.625\).

Key Concepts

Coin toss

The coin toss is a simple yet fundamental concept in probability. It involves flipping a standard two-sided coin to generate a random outcome. Each side of the coin, head or tail, has an equal likelihood of appearing. This means there are only two possible outcomes for any coin toss:

• Heads (H)
• Tails (T)

Each outcome has a probability of occurring that can be expressed as a fraction. The probability of flipping a head is \(\frac{1}{2}\), because there is 1 head and 2 possible outcomes. Similarly, the probability of flipping a tail is also \(\frac{1}{2}\). This fundamental concept of a coin toss helps illustrate the basis of probability, making it an essential tool for understanding more complex probability scenarios.

Deck of cards

A standard deck of cards contains 52 cards, and it offers a wide variety of outcomes for probability questions. Each card falls into one of four suits: spades, hearts, diamonds, and clubs. Each suit consists of 13 cards, including numbered cards from 2 to 10, as well as the face cards: Jack, Queen, King, and an Ace.

In probability exercises featuring a deck of cards, it is crucial to understand the composition of the deck. For example, if you're asked to determine the probability of drawing a club, you need to know:

• There are 13 clubs in the deck
• Thus, the probability of drawing a club is \(\frac{13}{52} = \frac{1}{4}\)

The understanding of these basics allows you to work through probability problems efficiently. Recognizing the structure of a deck is a powerful skill in solving card-related probability questions.

Addition rule for probability

The addition rule for probability is a technique used to calculate the probability of either of two events occurring. It's especially useful when dealing with scenarios where events can happen simultaneously. In the case of coin tosses and card draws, the goal is to find the likelihood of either event occurring.

To apply the addition rule, you use the formula:
\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]
where

• \(P(A)\) is the probability of the first event (e.g., tossing a head)
• \(P(B)\) is the probability of the second event (e.g., drawing a club)
• \(P(A \cap B)\) is the probability that both events happen at the same time

For the exercises using a coin toss and a deck of cards, this rule allows us to effectively calculate probabilities by adding the probabilities of each independent event, then subtracting any overlap where the events could simultaneously occur. Understanding this rule is instrumental for mastering various probability problems and achieving accurate results.