Problem 26 - Basic College Mathematics

Question

A coin is flipped twice. What is the probability that one head and one tail occur?

Step-by-Step Solution

Answer

The probability is \(\frac{1}{2}\).

Step 1: List All Possible Outcomes

When flipping a coin twice, list all the possible outcomes. Each flip has two possible outcomes: Head (H) or Tail (T). Therefore, the outcomes are: HH, HT, TH, and TT.

Step 2: Count Outcomes with One Head and One Tail

Identify the outcomes where there is one head and one tail. These outcomes are HT and TH.

Step 3: Calculate the Total Number of Outcomes

There are four possible outcomes in total: HH, HT, TH, and TT.

Step 4: Find the Probability

The probability is the number of favorable outcomes divided by the total number of outcomes. There are 2 favorable outcomes (HT and TH) and 4 total outcomes. The probability is therefore: \(\frac{2}{4} = \frac{1}{2}\)

Key Concepts

Outcomes

When we flip a coin, each flip results in an outcome. Since a coin has two sides, the possible outcomes for each flip are: Head (H) or Tail (T).
If we flip the coin twice, we need to consider all possible combinations of these individual outcomes.
These combinations, or total outcomes, can be listed as follows:

• HH (Head, Head)
• HT (Head, Tail)
• TH (Tail, Head)
• TT (Tail, Tail)

Each combination is an outcome, and listing all outcomes helps us understand all the possible results from a given scenario.

Favorable Outcomes

Favorable outcomes are the specific outcomes that satisfy the condition we are interested in. In this exercise, we want to determine the probability of getting one head and one tail when flipping the coin twice.
From the list of all possible outcomes—HH, HT, TH, TT—we identify which ones match our criteria:

• HT (Head, Tail)
• TH (Tail, Head)

These are the favorable outcomes because they consist of one head and one tail. Counting these favorable outcomes is crucial in calculating probability.

Total Outcomes

Total outcomes represent all the possible results that can occur when flipping the coin twice. There are four outcomes in total: HH, HT, TH, TT.
This number is essential because the probability of an event is calculated as the ratio of favorable outcomes to the total number of outcomes.
In this exercise, we determined there are 4 total outcomes.
By knowing both the total outcomes and the favorable outcomes, we can easily find the probability of a specific event occurring.