Problem 6 - Algebra 2 and Trigonometry

Question

In \(6-9,\) write an expression using sigma notation that can be used to find each probability. At least 10 heads when 15 coins are tossed

Step-by-Step Solution

Answer

\( (0.5)^{15} \sum_{k=10}^{15} \binom{15}{k} \)

Step 1: Identify the Problem

The task is to calculate the probability of getting at least 10 heads when tossing 15 coins. This is a binomial probability problem where we need to sum the probabilities of getting 10, 11, 12, 13, 14, and 15 heads.

Step 2: Understand Binomial Probability

For binomial probability, the probability of getting exactly 'k' successes (heads) in 'n' trials (coin tosses) is given by the formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where \( \binom{n}{k} \) is the binomial coefficient, \( p \) is the probability of success (0.5 for a fair coin), and \( n-k \) is the number of failures.

Step 3: Set Up Sigma Notation

We need to sum the binomial probabilities from 10 to 15 heads. This can be expressed using sigma notation as:\[ \sum_{k=10}^{15} \binom{15}{k} (0.5)^k (0.5)^{15-k} \] which simplifies to:\[ \sum_{k=10}^{15} \binom{15}{k} (0.5)^{15} \]

Step 4: Simplify the Expression

Combine the constants \( (0.5)^k \) and \( (0.5)^{15-k} \) inside the sigma expression to further simplify. Since \( (0.5)^k(0.5)^{15-k} = (0.5)^{15} \), the term \( (0.5)^{15} \) can be factored out:\[ (0.5)^{15} \sum_{k=10}^{15} \binom{15}{k} \]

Key Concepts

Sigma Notation

Sigma notation is a mathematical symbol used to represent the sum of a series. The symbol \( \Sigma \) stands for the summation of terms, usually expressed as \( \sum_{i=a}^{b} \) where \( a \) is the starting index and \( b \) is the ending index. In the context of binomial probability, sigma notation allows us to neatly express the sum of probabilities for multiple outcomes. In our problem, we want to find the probability of getting between 10 and 15 heads inclusive when tossing 15 coins. Instead of calculating each probability separately, sigma notation helps us write this as: \[\sum_{k=10}^{15} \binom{15}{k} (0.5)^{15}\]This effectively sums the probabilities of getting each number of heads from 10 through 15 in one compact expression.

Binomial Coefficient

The binomial coefficient, denoted as \( \binom{n}{k} \), is a fundamental part of combinatorial mathematics. It represents the number of ways to choose \( k \) successes from \( n \) trials. In simpler terms, it's how many ways you can pick \( k \) objects out of \( n \) without regard to order.

• \( n \) is the total number of trials or objects.
• \( k \) is the number of successes or objects to choose.

The formula for the binomial coefficient is \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \(!\) represents a factorial, the product of all positive integers up to that number. This coefficient is crucial for calculating each term in a binomial distribution, as it tells us how many ways each specific outcome—like getting 10 heads in 15 coin flips—can occur.

Probability of Success

In a binomial experiment, the probability of success \( p \) means the chance of achieving the desired outcome in a single trial. For a fair coin toss, which is our scenario, the probability of success is 0.5. Here, success is defined as the coin landing on heads. The entire concept of binomial probability hinges on this probability of success because it affects the likelihood of getting a specific number of successes over several trials. Each trial is independent, meaning the outcome of one doesn’t affect another. In the formula \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), \( p^k \) represents the probability of \( k \) successes happening, while \( (1-p)^{n-k} \) accounts for the remaining failures. This makes it a perfect model for situations like repeated coin tosses.

Fair Coin Toss

A fair coin toss is one where the coin is balanced, resulting in an equal probability of landing heads or tails—each has a probability of 0.5. In statistical terms, a fair coin toss is an example of a Bernoulli trial, an experiment with exactly two outcomes—success or failure (heads or tails).

• The concept of a fair coin is crucial when using binomial probability as it maintains the consistency of the outcomes.
• Since the tosses are independent, every flip is not affected by the previous results, maintaining the integrity of the calculations.

Using a fair coin in probability calculations ensures the accuracy and fairness of the outcomes, making it the bedrock of binomial probability problems like the one at hand: tossing a coin 15 times and calculating the likelihood of achieving at least 10 heads.