Problem 21 - College Algebra with Modeling and Visualization

Question

Find the probability of the compound event. Tossing a coin twice with the outcomes of two tails

Step-by-Step Solution

Answer

The probability of getting two tails in two coin tosses is \( \frac{1}{4} \) or 0.25.

Step 1: Identify the Sample Space

Since we are tossing a coin twice, each toss has 2 possible outcomes: heads (H) or tails (T). The possible outcomes for two tosses are: HH, HT, TH, TT. This gives a total of 4 outcomes in the sample space.

Step 2: Identify the Favorable Outcomes

We are looking for the probability of getting two tails, which is represented by the outcome TT. This is a single favorable outcome in the sample space.

Step 3: Use the Probability Formula

The probability of an event is given by the formula \( \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes in the sample space}} \). Here, the number of favorable outcomes for two tails is 1, and the total number of outcomes is 4.

Step 4: Calculate the Probability

Apply the numbers to the formula: \( \text{Probability of two tails} = \frac{1}{4} \). This results in a probability of \( \frac{1}{4} \) or 0.25.

Key Concepts

Sample Space

In probability theory, a sample space is a crucial concept as it encompasses all possible outcomes of a particular experiment. For the event of tossing a coin twice, the sample space consists of every possible combination of Heads (H) and Tails (T) that can result from these tosses. By listing each event, we consider:

• Two Heads (HH)
• Head then Tail (HT)
• Tail then Head (TH)
• Two Tails (TT)

This makes up our complete sample space of four possible outcomes. Understanding the sample space ensures that we are accounting for all potential results of the experiment, making our probability calculations accurate.

Favorable Outcomes

To find the probability of a specific event, we need to identify "favorable outcomes" - those outcomes that satisfy the criteria we're interested in. In the case of tossing a coin twice and looking for two tails, the favorable outcome is simply the combination TT. It stands alone in our sample space, making it unique among the four possible outcomes.
Recognizing favorable outcomes involves:

• Defining the event you're interested in
• Identifying which scenarios in the sample space meet this definition

Here, the event is getting two tails, with TT as the only favorable outcome. Clearly identifying these outcomes helps us proceed to calculate the probability accurately.

Probability Formula

The probability of an event gives us a measure of how likely that event is to occur. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes in the sample space. Mathematically, it is expressed as:
\[\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes in the sample space}} \]
Using our coin toss example, there is only 1 favorable outcome (TT) for getting two tails, out of the 4 possible outcomes.
Therefore, the probability is computed as:
\[\text{Probability of two tails} = \frac{1}{4} = 0.25\]This means there is a 25% chance of tossing two tails in this experiment. Understanding and applying the probability formula allows us to make quantitative predictions about future events based on past observations.