Problem 17 - College Algebra

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: Define the Sample Space

When tossing two coins simultaneously, the sample space includes all possible outcomes: heads on both (HH), heads on the first and tails on the second (HT), tails on the first and heads on the second (TH), and tails on both (TT). Thus, the sample space is {HH, HT, TH, TT}.

Step 2: Identify Favorable Outcomes

To find the probability of getting at least one tail, we consider all outcomes that include a tail. The favorable outcomes are {HT, TH, TT} because each contains at least one tail.

Step 3: Count the Number of Outcomes

Count the total number of possible outcomes in the sample space, which is 4 (i.e., HH, HT, TH, TT). Also, count the number of favorable outcomes, which is 3 (i.e., HT, TH, TT).

Step 4: Calculate the Probability

Use the formula for probability: \( P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \). Substitute the values: \( P(\text{at least one tail}) = \frac{3}{4} \).

Key Concepts

Sample Space

When we talk about the sample space in probability, we're discussing all the possible outcomes of a particular experiment or event. For example, if you toss two coins, like in our exercise, the experiment could result in several combinations of heads and tails. The sample space includes every possible outcome that can occur.
For two coins, the sample space can be written as:

• HH (both coins show heads)
• HT (the first coin shows heads and the second shows tails)
• TH (the first coin shows tails and the second shows heads)
• TT (both coins show tails)

This result means the sample space is \(\{HH, HT, TH, TT\}\). Understanding the sample space is a crucial fundamental step in solving any probability problem as it helps us determine what outcomes can realistically occur.

Favorable Outcomes

While the sample space encompasses all possible outcomes, favorable outcomes are those specific outcomes that satisfy the condition of the probability problem. In our exercise, the condition is that at least one coin should show tails when two coins are tossed. To identify the favorable outcomes, we scan through the sample space to find all combinations that meet this requirement:

• HT (heads on the first coin, tails on the second)
• TH (tails on the first coin, heads on the second)
• TT (tails on both coins)

These scenarios, HT, TH, and TT, are our favorable outcomes. Thus, we have a set of favorable outcomes \(\{HT, TH, TT\}\). It's these results in the sample space that guide us towards calculating the probability of the event occurring.

Probability Calculation

Probability measures how likely an event is to occur. To calculate it, we use a simple formula that compares the number of favorable outcomes with the total number of possible outcomes in the sample space.
The formula is: \[P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\]Here, we substitute the values: there are 3 favorable outcomes \(\{HT, TH, TT\}\) and the total number of possible outcomes is 4 \(\{HH, HT, TH, TT\}\). So, the probability of getting at least one tail when tossing two coins is: \[P(\text{at least one tail}) = \frac{3}{4}\]Thus, there is a 75% chance that you'll get at least one tail when you toss two coins. Having a solid understanding of this calculation helps in solving many real-world probability problems effectively.