Problem 1 - Algebra for College Students

Question

Two coins are tossed. Find the probabil- ity of tossing each of the following events: One head and one tail See below

Step-by-Step Solution

Answer

The probability of getting one head and one tail is \(\frac{1}{2}\).

Step 1: Identify the Sample Space

When two coins are tossed, each coin can result in two possible outcomes: heads (H) or tails (T). Therefore, the sample space for tossing two coins is: \(\{HH, HT, TH, TT\}\). This means there are 4 possible outcomes.

Step 2: Define the Event

The event of interest is getting one head and one tail. This could happen in two possible ways: getting a head on the first coin and a tail on the second coin (HT), or getting a tail on the first coin and a head on the second coin (TH). Thus, the event can be represented as: \(\{HT, TH\}\).

Step 3: Count the Number of Successful Outcomes

From the sample space \(\{HH, HT, TH, TT\}\), the outcomes that are in our event of interest are \(\{HT, TH\}\). There are 2 such outcomes.

Step 4: Calculate the Probability

The probability of an event is given by the formula \( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \). Here, the number of favorable outcomes is 2, and the total number of possible outcomes is 4. Thus, the probability is: \[ P(E) = \frac{2}{4} = \frac{1}{2} \]

Key Concepts

Sample Space

In probability, the sample space is the set of all possible outcomes of an experiment. For the exercise involving tossing two coins, each coin can show either heads (H) or tails (T). The sample space is noted as \( \{HH, HT, TH, TT\} \).

This means that there are 4 different outcomes when two coins are tossed:

• HH: Both coins show heads
• HT: The first coin shows heads, and the second coin shows tails
• TH: The first coin shows tails, and the second coin shows heads
• TT: Both coins show tails

Understanding the sample space is vital as it provides the foundation for calculating probabilities of events.

Events

Events in probability are defined as a particular set of outcomes from the sample space. They are what we are interested in observing out of all possible outcomes.

In the exercise, the event we are interested in is getting one head and one tail when tossing two coins. This can occur in two ways:

• HT: Head on the first coin and tail on the second coin
• TH: Tail on the first coin and head on the second coin

Hence, the event can be symbolized by the set \( \{HT, TH\} \). Recognizing which outcomes belong to your event helps narrow down what you need to focus on when calculating probability.

Outcomes

Outcomes are the individual results that can result from an experiment. For tossing two coins, each layout of heads and tails represents a unique outcome.

In our example, each sequence such as HH, HT, TH, and TT is an outcome. Specifically for the event "one head and one tail," the relevant outcomes are HT and TH. Identifying these outcomes allows you to count how many ways the desired event can happen, which is crucial for probability calculations.

Probability Calculation

Calculating probability involves determining how likely an event is to happen, based on the sample space and the event's outcomes.
The probability \( P(E) \) of an event \( E \) is calculated using the formula:

\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]
In the given exercise, there are 2 favorable outcomes (HT and TH) and a total of 4 possible outcomes in the sample space. Thus, the probability of getting one head and one tail is:

\[ P(E) = \frac{2}{4} = \frac{1}{2} \]
This means there is a 50% chance of tossing one head and one tail when two coins are flipped.