Problem 21 - Algebra 2

Question

Probability A coin is tossed ten times. The probability of heads on each toss is \(0.5 .\) Evaluate each probability. a. exactly 5 heads \(\quad\) b. exactly 6 heads \(\quad\) c. exactly 7 heads

Step-by-Step Solution

Answer

The probability of getting exactly a) 5 heads is calculated in step 1; b) 6 heads is calculated in step 2; c) 7 heads is calculated in step 3. Each step provides the respective probability.

Step 1: Evaluating probability for exactly 5 heads

Substitute \(k = 5, n = 10, p = q = 0.5\) into the formula and calculate the probability: \[ P(5;10,0.5) = C(10, 5) \cdot (0.5^5) \cdot ((1-0.5)^{10-5}) \]

Step 2: Evaluating probability for exactly 6 heads

Substitute \(k = 6, n = 10, p = q = 0.5\) into the formula and calculate the probability: \[ P(6;10,0.5) = C(10, 6) \cdot (0.5^6) \cdot ((1-0.5)^{10-6}) \]

Step 3: Evaluating probability for exactly 7 heads

Substitute \(k = 7, n = 10, p = q = 0.5\) into the formula and calculate the probability: \[ P(7;10,0.5) = C(10, 7) \cdot (0.5^7) \cdot ((1-0.5)^{10-7}) \]

Key Concepts

Probability Theory

Probability theory is a branch of mathematics that deals with the likelihood of different outcomes. In simple terms, it helps us to understand how probable something is to happen. If you think about any random event, such as a coin toss, this theory provides a structured way to calculate how often we can expect a certain result.

When you toss a fair coin, there are two possible outcomes: heads or tails. Each outcome has a probability of 0.5 because the coin is fair and balanced. This idea of assigning numbers to uncertain events is at the heart of probability theory. In more complex situations, like the one in our problem where the coin is tossed ten times, probability theory allows us to calculate the likelihood of getting a specific number of heads.

• Probability ranges from 0 to 1: 0 means an event will not happen, and 1 means it certainly will.
• The sum of probabilities for all possible outcomes in an experiment is always 1.
• Binomial probability, like in the exercise, involves experiments where there are two discrete outcomes, often referred to as success and failure.

Understanding these basic elements helps us analyze more complicated problems involving chance, such as predicting outcomes of sporting events or estimating risks in finance.

Combinatorics

Combinatorics is all about counting how many ways things can be organized or arranged, which is crucial in computing probabilities in experiments like our coin toss. In our example, we want to determine the number of ways we can get a specific number of heads when tossing a coin 10 times.

A significant tool used in combinatorics is the binomial coefficient, often represented as \( C(n, k) \) or "n choose k," which calculates the number of ways to choose \( k \) successes (e.g., heads) out of \( n \) trials (e.g., tosses). This coefficient is calculated as:
\[ C(n, k) = \frac{n!}{k!(n-k)!} \]

• \( n!\) ("n factorial") means you multiply all positive integers from 1 to \( n \).
• Combinatorics helps consider all possible arrangements to assess probabilities properly.
• In our coin toss experiment, combinatorics helps us calculate the number of ways to achieve a specific outcome, like getting exactly 5, 6, or 7 heads.

By mastering these counting techniques, you can solve a wide range of problems beyond just flipping coins, from arranging books on a shelf to complex statistical calculations.

Coin Toss Experiment

The coin toss experiment is a classic example in probability that serves as a perfect introduction to understanding probabilistic events. Tossing a coin is one of the simplest types of experiments because it involves only two outcomes: heads or tails. When you expand this simple experiment to multiple tosses, it becomes more complex and interesting because you can apply probability and combinatorial principles to assess outcomes.

In the exercise, we are asked to calculate the probability of obtaining a certain number of heads when a coin is tossed 10 times. This situation is modeled by a binomial distribution because:

• There are a fixed number of trials (10 tosses).
• Each trial has two possible outcomes (heads or tails).
• The probability of success (getting heads) is the same on every trial.

The probability formula we are using involves multiplying the number of ways to achieve the desired outcome by the probability of the outcome of each individual trial raised to the power of the number of successes and failures. Such experiments are foundational tools, and are applicable in real-world scenarios, such as calculating the chances of winning a game or evaluating possible outcomes of experiments in a controlled lab environment.