Problem 21 - Technical Mathematics with Calculus

Question

What is the probability of tossing 7 heads in 10 tosses of a fair coin?

Step-by-Step Solution

Answer

The probability of tossing exactly 7 heads in 10 tosses of a fair coin is 0.1171875.

Step 1: Understanding the Problem

This problem is an example of a binomial distribution, where there are two possible outcomes for each trial (tossing a head or a tail), and we want to find the probability of getting exactly 7 heads in 10 trials (tosses).

Step 2: Computing Individual Trial Probability

Since the coin is fair, the probability of getting a head (success) on any single toss is 0.5. This is because there are 2 possible outcomes, and each of them is equally likely.

Step 3: Using the Binomial Probability Formula

The binomial probability formula is given by: \( P(X = k) = {n \choose k} p^k (1-p)^{n-k} \), where \( n \) is the number of trials, \( k \) is the number of successes we are looking for, \( p \) is the probability of success on a single trial, and \( {n \choose k} \) is the binomial coefficient. For this case, \( n=10, k=7, \) and \( p=0.5 \).

Step 4: Calculating the Binomial Coefficient

The binomial coefficient \( {n \choose k} \) represents the number of ways to choose k successes out of n trials, and is calculated as \( {n \choose k} = \frac{n!}{k!(n-k)!} \). Substituting the values we get \( {10 \choose 7} = \frac{10!}{7!3!} = 120 \).

Step 5: Applying the Values to the Formula

Now apply the values to the binomial probability formula to get the probability: \( P(X = 7) = {10 \choose 7} \times 0.5^7 \times (1-0.5)^{10-7}\).

Step 6: Solving the Probability

Finally, calculate the probability: \(P(X=7) = 120 \times 0.5^7 \times 0.5^3 = 120 \times 0.5^{10} = 120 \times 0.0009765625 = 0.1171875\).

Key Concepts

Probability Theory

Probability theory is the branch of mathematics concerned with analyzing random events and quantifying the likelihood of different outcomes. In the context of a fair coin flip, the results (heads or tails) are random. Yet, we can predict the statistical frequency of these outcomes over a large number of trials based on probability principles.

At its core, probability values range from 0 to 1, where 0 indicates an impossibility and 1 represents a certainty. The probability of flipping a head with a fair coin is exactly 0.5, which means there is an equal chance for heads or tails. In the given exercise, we examine the event of flipping a coin 10 times and finding the probability of getting precisely 7 heads, applying principles of probability theory to solve the problem.

Binomial Coefficient

The binomial coefficient is a fundamental concept in combinatorics, representing the number of ways to choose a subset of items from a larger set, without regard to order. It is denoted as \( {n \choose k} \), where \( n \) is the total number of items and \( k \) is the number of items to choose.

Calculated using the factorials of the numbers involved (\( n! \) is the factorial of \( n \) and stands for the product of all positive integers up to \( n \)), the binomial coefficient formula is \( {n \choose k} = \frac{n!}{k!(n-k)!} \). In our exercise, to figure out the number of ways to get 7 heads out of 10 tosses, the binomial coefficient \( {10 \choose 7} \) reveals there are 120 different combinations where 7 heads could occur.

Binomial Distribution Formula

The binomial distribution formula is a blueprint for determining the probability of achieving a certain number of successes in a fixed number of trials, with just two possible outcomes (success or failure) and a constant probability of success in each trial. The formula, expressed as \( P(X = k) = {n \choose k} p^k (1-p)^{n-k} \), combines the concepts of individual trial probability (\( p \) for success), the binomial coefficient (\( {n \choose k} \)), the number of successes (\( k \) heads), and the number of trials (\( n \)) for a complete probability calculation.

In the context of our exercise seeking the probability of flipping 7 heads in 10 coin tosses, we observed the formula at work with a balanced probability (\( p = 0.5 \) for heads) and applied it to evaluate the precise chance of this specific outcome. Ultimately, it quantified the likelihood to be approximately 0.1172, meaning there is close to a 12% chance of flipping exactly 7 heads in 10 tosses.