Problem 16 - Algebra 2

Question

Find each probability if a die is rolled 4 times. \(P(\text { exactly three } 3 \mathrm{s})\)

Step-by-Step Solution

Answer

The probability is \( \frac{5}{324} \).

Step 1: Understand the problem

We need to find the probability of rolling exactly three 3's when a die is rolled four times. This is a classic example of a binomial probability problem where each trial is independent, and we have a fixed number of trials with the same probability of success.

Step 2: Set up the binomial probability formula

The formula for the binomial probability is given by: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where \( n \) is the number of trials, \( k \) is the number of successful trials, and \( p \) is the probability of success on a single trial. In this case, \( n = 4 \), \( k = 3 \), and \( p = \frac{1}{6} \) since a die has six faces.

Step 3: Calculate the probability of success and failure

The probability of rolling a 3 is \( p = \frac{1}{6} \), and the probability of not rolling a 3 is \( 1 - p = \frac{5}{6} \).

Step 4: Calculate the binomial coefficient

Calculate \( \binom{4}{3} \), which represents the number of ways to choose 3 successes (rolling a 3) in 4 trials. \[ \binom{4}{3} = \frac{4!}{3!(4-3)!} = 4 \]

Step 5: Apply the binomial probability formula

Substitute the values into the binomial probability formula. \[ P(X = 3) = \binom{4}{3} \left(\frac{1}{6}\right)^3 \left(\frac{5}{6}\right)^{4-3} \] \[ P(X = 3) = 4 \times \left(\frac{1}{6}\right)^3 \times \frac{5}{6} \]

Step 6: Calculate the final result

Compute the expression. \[ P(X = 3) = 4 \times \frac{1}{216} \times \frac{5}{6} = 4 \times \frac{5}{1296} = \frac{20}{1296} = \frac{5}{324} \]

Step 7: Simplify the probability

The probability \( P(\text{exactly three } 3\text{s}) \) simplifies to \( \frac{5}{324} \).

Key Concepts

Probability of Success

In binomial probability, the probability of success (\(p\)) represents the likelihood of achieving a specific outcome in an independent trial. In our dice example, the probability of rolling a three on a single roll is a success. Since there are six faces on a die and each face is equally likely to appear, the probability of rolling any particular number, such as a three, is \(\frac{1}{6}\). This means, every time we roll the die, there’s a \(\frac{1}{6}\) chance we get a three. It's crucial to understand that this probability remains constant across all trials, making each roll an independent event. Remember, success doesn't mean winning or achieving a goal; it simply refers to the event we are interested in observing.

Binomial Coefficient

The binomial coefficient is a key component in calculating binomial probabilities. It is denoted as \(\binom{n}{k}\), where \(n\) is the total number of trials, and \(k\) is the number of successful trials we are interested in. Mathematically, it's calculated using the formula:- \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]This formula counts the number of ways to select \(k\) successes from \(n\) trials. In the context of our example, \(\binom{4}{3} = 4\) since there are four ways to roll exactly three threes in four dice rolls. This step is essential as it tailors the probability calculation to the specific outcome we are examining.

Independent Trials

Independent trials are a fundamental aspect of binomial probability. Independence means that the outcome of one trial does not affect the outcome of another. For example, rolling a die four times involves independent trials because the result of one roll doesn't impact the next roll. This independence is crucial because it allows us to apply the binomial probability formula without considering complex interdependencies between trials. Since the probability of success remains the same for each trial, we can focus on computing the number of successes over the total trials without worrying about the influence of previous outcomes.

Binomial Probability Formula

The binomial probability formula is a mathematical tool used to find the probability of a given number of successes in a fixed number of independent trials. The formula is expressed as:- \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]Where:- \(n\) is the number of trials- \(k\) is the number of successes- \(p\) is the probability of success in each trial- \((1-p)\) is the probability of failureUtilizing all values in the formula lets us determine the probability of achieving exactly \(k\) successes after \(n\) trials. For example, in our scenario (\(n = 4\), \(k = 3\), \(p = \frac{1}{6}\)), the formula helps us compute the precise likelihood of rolling exactly three threes in four dice rolls. This comprehensive approach is what makes the formula indispensable for solving binomial probability problems.