Problem 25 - Basic College Mathematics

Question

A coin is flipped twice. What is the probability that two heads occur?

Step-by-Step Solution

Answer

The probability of flipping two heads is \( \frac{1}{4} \).

Step 1: Determine the Sample Space

Identify all possible outcomes when a coin is flipped twice. The outcomes are: HH, HT, TH, TT, where 'H' stands for heads and 'T' stands for tails.

Step 2: Calculate the Total Number of Outcomes

Count the total number of possible outcomes. There are 4 possible outcomes: HH, HT, TH, and TT.

Step 3: Identify the Favorable Outcomes

Determine which outcomes involve flipping two heads. There is only one such outcome: HH.

Step 4: Calculate the Probability

Use the probability formula: \( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \). Here, the number of favorable outcomes is 1 and the total number of outcomes is 4. Hence, the probability is: \( P(\text{Two heads}) = \frac{1}{4} \).

Key Concepts

Sample Space

In probability theory, understanding the sample space is crucial. The sample space encompasses all the possible outcomes that can occur in an experiment. When flipping a coin twice, each flip can result in either heads (H) or tails (T).

Combining these possibilities, we get the following outcomes:

• HH (both flips landing heads)
• HT (first flip heads, second flip tails)
• TH (first flip tails, second flip heads)
• TT (both flips landing tails)

This list represents our sample space for the experiment of flipping a coin twice.

Recognizing the sample space is foundational because it sets the stage for determining probabilities.

Favorable Outcomes

After establishing the sample space, the next step is to identify the favorable outcomes. Favorable outcomes are those specific results that satisfy the condition of our probability question.

In the given problem, we are interested in finding the probability of flipping two heads. From our sample space (HH, HT, TH, TT), we see that only one outcome meets this criterion: HH.

This single outcome (HH) out of the four possible scenarios is what we call a favorable outcome for our experiment.

Probability Formula

Calculating probability involves the probability formula, a key tool in probability theory. The formula is expressed as:
\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]
This formula helps us quantify the likelihood of an event occurring. Let's break it down:

• Numerator (top part of the fraction): This represents the number of outcomes that are favorable to our event. In our problem, that’s the number of outcomes with two heads.
• Denominator (bottom part of the fraction): This is the total number of outcomes in the sample space. For our problem, it’s the total number of outcomes when flipping a coin twice.

Using this formula makes it easier to solve probability questions by following a systematic approach.

Total Outcomes

Understanding the concept of total outcomes is vital for solving probability problems. The total outcomes are simply all the possible results we can get from an experiment. Let's take our coin flipping example:

We've already listed the sample space, which comprises HH, HT, TH, and TT. This sample space represents all four total outcomes when a coin is flipped twice.

Recognizing the total number of outcomes helps in accurately applying the probability formula. Here, the total number of outcomes is 4.

By knowing both the total outcomes and favorable outcomes, you can precisely compute the probability of any event within the experiment.