Problem 9 - College Algebra with Modeling and Visualization

Question

Find the probability of each event. Tossing a head with a fair coin

Step-by-Step Solution

Answer

The probability of tossing a head with a fair coin is \( \frac{1}{2} \).

Step 1: Understand the Problem

We are asked to find the probability of getting a head when tossing a fair coin. A fair coin means that the likelihood of getting a head is the same as getting a tail.

Step 2: Identify the Possible Outcomes

When tossing a fair coin, there are two possible outcomes: either heads (H) or tails (T).

Step 3: Determine the Probability Formula

Probability of an event is given by the formula: \( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \).

Step 4: Apply the Probability Formula

To find the probability of getting a head (H), use the formula: \( P(H) = \frac{1}{2} \), since there is 1 favorable outcome (heads) and 2 possible outcomes (heads or tails).

Key Concepts

Fair Coin

A fair coin is a concept fundamental to understanding probability in scenarios involving coin tosses. When we say a coin is "fair," it simply means that it has an equal chance of landing on heads as it does on tails.
This property of fairness is crucial when calculating probabilities because it assures uniform randomness in outcomes.
Here are a few key points about a fair coin:

• Each side of the coin is equally weighted and shaped, ensuring no bias towards one side landing more than the other.
• The symmetry of the coin is what provides the equality, so that no matter how many times we flip, heads and tails are equally likely.
• A fair coin is often used in probabilistic experiments to model binary outcomes because its predictability is straightforward.

Possible Outcomes

When analyzing the possible outcomes of a coin toss, it is important to understand what "outcomes" are. An outcome is simply the result of an experiment—in this case, a flip of a coin.
For a fair coin, there are two possible outcomes:

• Heads (H)
• Tails (T)

These outcomes are exhaustive and mutually exclusive, meaning one and only one can occur per toss.
The concept can be expanded beyond a single toss. For multiple tosses, the outcomes grow exponentially. For example, in two tosses, there are four possible outcomes: HH, HT, TH, and TT. This growth pattern is predictable and useful in more complex probability calculations.

Probability Formula

Understanding the probability formula is key to solving problems involving chance and prediction.
In probability theory, the likelihood of an event occurring is calculated using a simple formula:\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]This formula helps you determine how often you can expect a particular result when performing an experiment like a coin toss.
For example, if you are tossing a fair coin and want to find the probability of getting heads, you would:

• Count the number of favorable outcomes, which is 1 (just heads).
• Identify the total number of possible outcomes, which is 2 (heads and tails).

So, the probability of flipping heads, \[ P(H) = \frac{1}{2} \] This means there is a 50% chance of landing a head on any single toss. The probability formula is an essential tool for anyone looking to calculate chances in scenarios involving random events.