Problem 15 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The probability is \( \frac{125}{324} \).

Step 1: Understanding the Problem

We need to find the probability of rolling a 3 exactly once when rolling a die four times. Each roll is an independent event with a probability of rolling a 3 being \( \frac{1}{6} \).

Step 1: Calculating the Success Probability

The probability of rolling a 3 on one roll of a die is \( \frac{1}{6} \). Since we need exactly one 3, the probability of rolling a 3 on one out of four rolls is \( \left( \frac{1}{6} \right) \).

Step 2: Calculating the Failure Probability

The probability of not rolling a 3 on a single roll is \( 1 - \frac{1}{6} = \frac{5}{6} \). For the other three rolls (not rolling a 3), the probability is \( \left( \frac{5}{6} \right)^3 \).

Step 3: Applying the Binomial Probability Formula

The scenario fits a binomial probability model where \( n = 4 \) (the number of trials), \( k = 1 \) (the number of successful trials), \( p = \frac{1}{6} \) (the probability of success on a single trial), and \( 1-p = \frac{5}{6} \) (the probability of failure). The binomial probability formula is:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]Substituting in the values, we have:\[ P(X = 1) = \binom{4}{1} \left( \frac{1}{6} \right)^1 \left( \frac{5}{6} \right)^{3} \]

Step 4: Evaluating the Combination Part

Calculate the binomial coefficient \( \binom{4}{1} \), which is the number of ways to choose 1 successful roll out of 4:\[ \binom{4}{1} = 4 \]

Step 5: Evaluating the Probability Expression

Now evaluate the probability expression:\[ 4 \times \left( \frac{1}{6} \right) \times \left( \frac{5}{6} \right)^{3} = 4 \times \frac{1}{6} \times \frac{125}{216} \]Perform the multiplication:\[ \frac{500}{1296} \]

Step 6: Simplifying the Probability

Simplify \( \frac{500}{1296} \) by dividing both the numerator and the denominator by their greatest common divisor (4):\[ \frac{500 \div 4}{1296 \div 4} = \frac{125}{324} \]

Step 8: Conclusion

Thus, the probability of rolling exactly one 3 when rolling a die four times is \( \frac{125}{324} \).

Key Concepts

Binomial Probability

When dealing with the word "binomial," it indicates we are considering two possible outcomes in a scenario, often termed as "success" and "failure." Binomial probability specifically refers to situations where you have a series of independent experiments or trials, and you are interested in the number of successful outcomes.
In the context of rolling a die, rolling a 3 can represent a "success," while any other outcome is seen as a "failure."

To calculate binomial probability, we use the binomial probability formula:

• \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

The components are:

• \( n \): total number of trials or attempts
• \( k \): number of successes
• \( p \): probability of success in each trial
• \( 1-p \): probability of failure in each trial

This formula provides a straightforward method to find the probability of "k" successes in "n" trials.

Independent Events

Events are considered independent if the outcome of one event does not influence the outcome of another. When rolling a standard six-sided die, each roll is independent, meaning rolling a 3 in one trial does not change the probability of rolling a 3 in the next trial.
This independence is crucial in binomial probability because it allows us to multiply probabilities of individual outcomes to get the combined probability of a sequence of events.

• For example, the probability of rolling a 3 in one trial is \( \frac{1}{6} \).
• The probability of not rolling a 3 is \( \frac{5}{6} \).

Since these are independent, for four rolls, the probability calculations remain stable across each attempt.
This concept simplifies calculating cumulative probabilities over multiple trials.

Binomial Coefficient

The binomial coefficient, represented as \( \binom{n}{k} \), is crucial in determining the number of ways to arrange "k" successes in "n" trials.
Using the binomial coefficient, you can find out how many different ways you can achieve a desired outcome.

• In our rolling dice example, \( \binom{4}{1} \) helps determine the arrangements to roll exactly one 3 in four attempts.

This coefficient is calculated using combinations, focusing on selections regardless of order. The formula used is:

• \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]

Here, \( n! \) (n factorial) represents the product of all numbers up to \( n \).
It shows us the different patterns of achieving "k" successes in "n" attempts without considering the sequence.

Success and Failure Probability

In binomial probability, identifying the success and failure probabilities is foundational. Each scenario has clear parameters for what constitutes success and failure.

For a single die roll:

• "Success" (getting a 3) has a probability of \( \frac{1}{6} \).
• "Failure" (not getting a 3) has a probability of \( \frac{5}{6} \).

These probabilities remain constant, given each event is independent.
During calculations, success probability is raised to the power of the number of successes \( k \), and failure probability is raised to the number of failures \( n-k \).
These computations provide the exact numerical likelihood of achieving the desired number of successful outcomes.