Problem 17 - College Algebra with Corequisite Support

Question

For the following exercises, two coins are tossed. Find the probability of tossing at least one tail.

Step-by-Step Solution

Answer

The probability of tossing at least one tail is \( \frac{3}{4} \).

Step 1: List All Possible Outcomes

When two coins are tossed, the possible outcomes are: Head-Head (HH), Head-Tail (HT), Tail-Head (TH), and Tail-Tail (TT). In terms of probability, there are 4 possible equally likely outcomes in total.

Step 2: Identify Favorable Outcomes for At Least One Tail

To meet the condition of at least one tail, the favorable outcomes are: HT, TH, and TT. This gives us 3 outcomes where there is at least one tail.

Step 3: Calculate the Probability of At Least One Tail

The probability of an event is given by the number of favorable outcomes divided by the number of total possible outcomes. Thus, the probability of tossing at least one tail is given by \( \frac{3}{4} \).

Key Concepts

Coin Toss

Coin tossing is a common example used in probability due to its simplicity and ease of understanding. A coin has two sides: heads and tails. When you toss a coin, the result of the toss is one of these two outcomes.
When two coins are tossed, there are more possible outcomes than tossing one coin. Each coin has an independent and equal chance of landing on heads or tails.

• If you toss one coin, you get one of two outcomes: heads (H) or tails (T).
• Tossing two coins at the same time increases the outcomes as each toss is independent of the other.

Understanding the outcomes of a coin toss is foundational to grasping probability concepts.

Favorable Outcomes

Favorable outcomes are those outcomes that match the condition we are considering. For example, when you want to know the probability of getting at least one tail in two coin tosses, you focus on the outcomes that meet this requirement.
In the given problem, these outcomes are:

• HT (Head-Tail)
• TH (Tail-Head)
• TT (Tail-Tail)

These outcomes are termed favorable because they satisfy the condition of having at least one tail, which is what we are trying to find the probability for. Understanding which outcomes are favorable is crucial in calculating probabilities.

Sample Space

Sample space is the set of all possible outcomes of a random experiment. It's the complete list of all outcomes that can occur in an experiment.
For the coin toss problem:

• Sample Space for one coin = {H, T}
• Sample Space for two coins = {HH, HT, TH, TT}

This concept is important because the probability of an event is calculated with reference to the sample space. It helps you determine both the total number of possible outcomes and the number of favorable outcomes.

Equally Likely Outcomes

Equally likely outcomes are outcomes that have the same probability of occurring. In a fair coin toss, heads and tails are equally likely because the coin is unbiased.
In the case of two coins, all the possible outcomes (HH, HT, TH, TT) are equally likely, meaning they have the same chance of occurring.

• Probability of HH = Probability of HT = Probability of TH = Probability of TT = \( \frac{1}{4} \)

Knowing that outcomes are equally likely simplifies the process of calculating probabilities, as it allows you to assume uniform distribution across possible events. This assumption is key in determining the probability for complex random experiments.