Problem 20 - Precalculus with Limits

Question

In Exercises 15 - 20, find the probability for the experiment of tossing a coin three times. Use the sample space \( S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} \). The probability of getting at least two heads

Step-by-Step Solution

Answer

The probability of getting at least two heads when tossing a coin three times is 0.5

Step 1: Understand the Sample Space

The sample space is \( S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\} \). It represents all the possible outcomes obtained by tossing a coin three times. Each outcome is a string of three characters, where 'H' stands for Heads and 'T' for Tails.

Step 2: Identify Outcomes with At Least Two Heads

Next, go through the sample space and identify the outcomes in which there are at least two heads. These outcomes are: HHH, HHT, HTH, and THH. Thus, there are four such outcomes.

Step 3: Calculate the Probability

Probability is calculated as the ratio of the number of ways an event can occur to the total number of outcomes. In this case, the event is getting at least two heads. There are four such outcomes and eight total outcomes. Thus, the probability is \( \frac{4}{8} = 0.5 \)

Key Concepts

Sample Space

When learning about probability, the first step is to understand the concept of a sample space. The sample space, often denoted as \( S \), is the set of all possible outcomes of a random experiment. To illustrate, let's consider tossing a coin. If you toss a coin once, the sample space is \( S = \{H, T\} \), where 'H' represents heads and 'T' represents tails. However, when tossing a coin multiple times, the scenario becomes more interesting.

Imagine flipping a coin three times. Each flip is independent, meaning the result of one flip doesn't affect the next. The sample space for this experiment includes every possible combination of heads and tails over the three flips, giving us \( S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\} \). There are 8 outcomes in total, because each toss has 2 possible outcomes and there're 3 tosses, resulting in \(2^3\) or 8 total combinations. Understanding the sample space is crucial because it provides the foundation for calculating probabilities of different events.

Probability Calculation

Probability calculation is the process of determining the likelihood of a specific event occurring within the sample space. The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. To make this more digestible, consider our coin-tossing example.

If we want to know the probability of getting at least two heads in three coin tosses, we first identify the favorable outcomes. These are the outcomes from the sample space where at least two of the flips land on heads. Looking at our sample space, the favorable outcomes are HHH, HHT, HTH, and THH. There are four outcomes that satisfy our condition. Now, remember our total number of possible outcomes is eight. Hence, the probability of our event (at least two heads) is \( \frac{4}{8} \) or 0.5. Put simply, there's a 50% chance of tossing at least two heads in three tosses.

Combinatorics

Combinatorics plays a critical role in calculating probabilities, especially when dealing with complex experiments. It's the area of mathematics concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It provides us with tools to count the combinations and permutations of a given set without having to list them all out.

In our three coin tosses example, combinatorics is useful for understanding how the 8 different outcomes of the sample space are derived. The number of possible outcomes is based on the principle that each coin toss has 2 outcomes, and the number of total outcomes is the product of the possibilities for each individual event — hence \(2 \times 2 \times 2 = 8\). For more complex events, such as the number of ways to arrange a group of people or select a committee, combinatorial formulas and theorems are used to simplify calculations and understand the structure of the sample space. Familiarity with combinatorial concepts is essential for a deeper understanding of probability and its applications.