Problem 2 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the probability of the given event. The coin lands heads exactly once.

Step-by-Step Solution

Answer

The probability of the coin landing heads exactly once in one flip is \(\frac{1}{2}\), or 50%.

Step 1: Identify the given information

In this problem, we are given: n = 1 (since we are concerned about only one coin flip), k = 1 (we want the coin to land heads exactly once), and p = 1/2 (the probability of getting heads in each flip).

Step 2: Calculate C(n, k)

Next, we need to calculate the number of combinations of n objects taken k at a time (C(n, k)): C(n, k) = n! / (k! * (n-k)!) In this case: C(1, 1) = 1! / (1! * (1-1)!) = 1 / (1 * 1) = 1

Step 3: Apply the binomial probability formula

Now that we have C(n, k), we can use the binomial probability formula: P(X = k) = C(n, k) * p^k * (1 - p)^(n-k) Substitute the values: P(X = 1) = C(1, 1) * (1/2)^1 * (1 - 1/2)^(1-1) P(X = 1) = 1 * (1/2) * (1/2)^0 P(X = 1) = 1 * (1/2) * 1 P(X = 1) = 1/2

Step 4: Interpret the result

The probability of the coin landing heads exactly once in one flip is 1/2, or 50%.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and finding patterns in sets of objects. When dealing with problems that involve combinations, like flipping a coin, we often want to determine how many ways a certain event can occur. For instance, when calculating combinations, we use the formula \( C(n, k) = \frac{n!}{k!(n-k)!} \). This formula determines the number of ways to choose \( k \) successes out of \( n \) trials.

In the given exercise, we calculated \( C(1, 1) \), which involved a simple choice. Since we are considering only one flip, the number of combinations is just 1: we can only choose one head out of one flip. This is a straightforward example showing how combinatorics help in understanding probabilities by counting possibilities. When problems grow more complex, with more flips or more variables, combinatorial tools remain crucial for accurate probability calculations.

Probability Theory

Probability theory is the mathematical framework that allows us to model random events and quantify uncertainty. In a coin flip, the probability of different outcomes, like heads or tails, is calculated based on the expected likelihood of those events. For a fair coin, each side has an equal probability of appearing, which is \( \frac{1}{2} \).

With problems involving binomial probability, such as flipping coins, the probabilities of multiple events can be combined using the binomial probability formula. This is given by \( P(X = k) = C(n, k) \cdot p^k \cdot (1 - p)^{(n-k)} \), where \( k \) is the number of successful outcomes, \( n \) is the total number of trials, and \( p \) is the probability of a single successful outcome (here, getting heads).

In this exercise, the concern is with the single event of a coin landing heads in one flip, which has a probability of \( 1/2 \). Probability theory gives us this simple yet powerful way to describe and compute the likelihood of outcomes, helping us make predictions and assessments about seemingly random events.

Mathematical Statistics

Mathematical statistics involves the application of probability theory to analyze and draw inferences about data or random processes. It is the foundation of many techniques used for statistically analyzing experiments and real-world data.

In contexts involving binomial events like coin flips, statistics helps us understand the variation and distribution of results over repeated trials. For a single coin flip, mathematical statistics tells us that the expected proportion of heads should be \( 0.5 \), matching our earlier probability result. If we were to flip a coin many times, mathematical statistics tools would help us analyze the outcomes systematically and compare them with our theoretical predictions.

This exercise with a single flip provides a foundational understanding of probability processes and sets the stage for more complex statistical exploration. With more flips, we might use statistics to model the distribution of results, compute confidence intervals, or perform hypothesis testing to assess our assumptions about the coin's fairness.