Problem 22 - College Algebra with Modeling and Visualization

Question

Find the probability of the compound event. Tossing a coin three times with the outcomes of three heads

Step-by-Step Solution

Answer

The probability of tossing three heads is \( \frac{1}{8} \).

Step 1: Understanding Outcomes for a Single Toss

For a single toss of a fair coin, there are 2 possible outcomes: heads (H) or tails (T).

Step 2: Determine Total Number of Outcomes for Three Tosses

When a coin is tossed three times, each toss is independent, and each has 2 outcomes. Therefore, the total number of outcomes for three tosses is calculated as:\[ 2 \times 2 \times 2 = 2^3 = 8 \].

Step 3: Identifying the Favorable Outcomes

The favorable outcome for three heads is HHH (heads on the first, second, and third toss). This is one specific sequence among all possible outcomes.

Step 4: Calculating Probability of Three Heads

The probability of an event is given by the number of favorable outcomes divided by the total number of outcomes. Therefore, the probability of getting three heads is:\[ \frac{1}{8} \].

Key Concepts

Compound Events

A compound event refers to any event that involves the combination of two or more simple events. For example, when tossing a coin multiple times, each individual toss is a simple event, but considering the results of all tosses together forms a compound event. To better understand, imagine throwing a coin three times. Each outcome—whether heads or tails—is a simple event. The compound event would be the collection of all possible sequences of heads and tails obtained from these tosses. Therefore, the compound event comprises multiple possibilities or sequences, such as HHH, HHT, THT, etc.
In probability terms, understanding compound events is crucial because it helps you calculate the probability for scenarios involving more than one action, like multiple coin tosses or rolling multiple dice.

Independent Events

Independent events are events where the outcome of one does not influence the outcome of another. In the context of tossing a coin, the result of one toss does not affect the result of any subsequent toss. This principle can be used to simplify the calculation of probabilities for compound events. When multiple events are independent, the probability of all occurring is the product of their individual probabilities:

• For example, the probability of landing heads on a single toss is \( \frac{1}{2} \).
• Therefore, for three independent tosses, the probability to get heads in all three events is \( \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \).

This multiplication rule only applies because each coin toss is independent of the others. It simplifies calculating probabilities for compound events, making the math manageable and precise.

Favorable Outcomes

Favorable outcomes are specific results within a set of possible outcomes that satisfy the condition of a given event. In probability, determining favorable outcomes is essential because it directly influences the calculation of an event's probability.Consider our exercise, where we're interested in the compound event of getting three heads when tossing a coin three times. The favorable outcome here is the sequence HHH, where each toss results in heads. This is just one possibility out of all the potential sequences:

• HHH
• HHT
• HTH
• THH
• and so on...

When calculating the probability of our event, we count only this one favorable outcome against the total number of all possible outcomes (in this case, 8). That's why the probability of obtaining three heads is \( \frac{1}{8} \). Understanding which outcomes are favorable allows us to apply probability formulas correctly, ensuring accurate problem-solving.