Problem 44 - A Graphical Approach to College Algebra

Question

A die is rolled 12 times. Approximate the probability of rolling the following. No more than 1 one

Step-by-Step Solution

Answer

The probability of rolling no more than 1 one is approximately 0.381.

Step 1: Define the Experiment

We are rolling a six-sided die 12 times. Let the event of interest be rolling a one on the die.

Step 2: Identify the Probability of Success

For a single roll of a die, the probability of rolling a one (success) is \(p = \frac{1}{6}\).

Step 3: Recognize the Distribution

The scenario describes a binomial distribution where the number of trials \(n = 12\), and the probability of success for each trial \(p = \frac{1}{6}\).

Step 4: Calculate Probability for 0 Ones

The probability of rolling zero ones is given by the formula for binomial probabilities: \[P(X = 0) = \binom{12}{0} \left(\frac{1}{6}\right)^0 \left(1 - \frac{1}{6}\right)^{12}\]Calculating gives:\[P(X = 0) = 1 \times 1 \times \left(\frac{5}{6}\right)^{12} = \left(\frac{5}{6}\right)^{12} \approx 0.112\]

Step 5: Calculate Probability for 1 One

Now, calculate the probability of rolling exactly one one:\[P(X = 1) = \binom{12}{1} \left(\frac{1}{6}\right)^1 \left(1 - \frac{1}{6}\right)^{11}\]Calculating gives:\[P(X = 1) = 12 \times \frac{1}{6} \times \left(\frac{5}{6}\right)^{11} \approx 0.269\]

Step 6: Add Probabilities

The probability of rolling no more than 1 one is the sum of the probabilities for rolling zero or one one:\[P(X \leq 1) = P(X = 0) + P(X = 1)\]\[P(X \leq 1) = 0.112 + 0.269 = 0.381\]

Key Concepts

Probability Theory

Probability theory is the branch of mathematics that deals with the likelihood of events occurring. It helps us understand and quantify uncertainty. In the context of rolling dice, probability theory allows us to calculate how likely certain outcomes are when we roll a die multiple times. For every possible event, such as rolling a one, we can determine the probability by using established formulas.

• Each outcome of a die roll is equally likely when the die is fair, meaning each face has a probability of \( \frac{1}{6} \).
• By understanding probability, we can calculate more complex probabilities like getting a specific number of ones in a series of die rolls.

Probability theory provides a systematic approach to compute these probabilities, often involving concepts like binomial distribution when events are independent and repeated a fixed number of times.

Random Experiments

A random experiment is any process or action that lead to one of several possible outcomes, where the outcome cannot be predicted with certainty. Rolling a die is a classic example of a random experiment. Each roll is independent of the previous ones, and there are six possible outcomes each time.

• Random experiments are essential for understanding random variables and their associated probabilities.
• In the case of rolling a die, the random experiment consists of 12 independent trials, each with 6 potential outcomes, where particularly rolling a one is our outcome of interest.

In our exercise, understanding the independence of each die roll is key to applying the binomial distribution, as each roll is a separate random experiment with consistent conditions.

Probability of Success

The probability of success in a random experiment refers to the chance of a particular event occurring. For a single roll of a die, the success probability of rolling a one is \( \frac{1}{6} \), since there are six sides on the die and only one side with a one.

• When dealing with multiple trials, like rolling the die 12 times, the concept extends to a binomial context where we want to know about successes across multiple attempts.
• This requires calculating if certain numbers of successes happen, such as rolling no more than one one, as seen in our exercise.

Understanding the probability of success in each trial helps in using the binomial distribution formula to find out the combined probabilities of different outcomes across multiple trials.