Problem 16 - College Algebra with Corequisite Support

Question

For the following exercises, two coins are tossed. Find the probability of tossing exactly one tail.

Step-by-Step Solution

Answer

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

Step 1: List Possible Outcomes

Two coins are tossed. Each coin can either land on Heads (H) or Tails (T). The possible outcomes for two coins are: HH, HT, TH, TT. This makes a total of 4 possible outcomes.

Step 2: Identify Favorable Outcomes

We want exactly one tail in our outcomes. The outcomes that have exactly one tail are HT and TH. So, there are 2 favorable outcomes.

Step 3: Calculate Probability

Probability is calculated as the number of favorable outcomes divided by the total number of possible outcomes. Thus, the probability of exactly one tail is \( \frac{2}{4} = \frac{1}{2} \).

Key Concepts

Outcomes

In the context of probability, an **outcome** refers to a possible result of a random event. When tossing coins, each flip results in one of two outcomes: **Heads (H)** or **Tails (T)**. When two coins are tossed together, we can think of this as a combined event consisting of two individual outcomes.

For two coins, these combined outcomes can be expressed as pairs:

• HH (both coins show Heads)
• HT (first coin shows Heads, second coin shows Tails)
• TH (first coin shows Tails, second coin shows Heads)
• TT (both coins show Tails)

These combinations represent all the possible results when flipping two coins. Since each coin toss is independent, the combinations can be predicted by multiplying the number of outcomes for each coin. Thus, with 2 outcomes per coin and 2 coins, we have a total of 4 outcomes.

Favorable outcomes

In probability, a **favorable outcome** is one that satisfies the condition we are interested in. For this exercise, we are specifically looking for the outcome where there is exactly one Tail when two coins are tossed.

From our list of possible outcomes **(HH, HT, TH, TT)**, we identify the outcomes that meet this condition:

• HT (first coin shows Heads, second coin shows Tails)
• TH (first coin shows Tails, second coin shows Heads)

These are the only outcomes in our list of four that include exactly one Tail. Therefore, we have 2 favorable outcomes. Understanding which outcomes are favorable is crucial in calculating probability, as it directly determines the numerator in our probability fraction.

Coin toss

A coin toss is a classic example of a simple random experiment in probability. Each flip of a coin results in one of two possible outcomes, known as **Heads (H)** or **Tails (T)**. This makes a coin toss a perfectly balanced or **fair** event, assuming the coin isn't biased.

When two coins are tossed simultaneously, each coin acts independently of the other. This independence implies that the outcome of one coin does not affect the other, allowing us to consider multiplicative rules for determining probabilities.

For example, since there are two possible outcomes per coin, and two coins, the total number of possible outcomes is calculated as:

• 2 outcomes for the first coin
• 2 outcomes for the second coin

Multiplying these gives us the 4 possible outcomes we discussed earlier: HH, HT, TH, TT. Coin tosses are foundational for understanding more complex probability concepts due to their simplicity and independence properties.