Problem 51 - Algebra 2

Question

A die is rolled three times. Find each probability. \(P(\text { three even numbers) }\)

Step-by-Step Solution

Answer

The probability of rolling three even numbers is \( \frac{1}{8} \).

Step 1: Identify even numbers on a die

A standard six-sided die has the numbers 1 through 6. The even numbers on a die are 2, 4, and 6. Therefore, there are 3 even numbers.

Step 2: Calculate the probability of rolling an even number

Since there are 3 even numbers and 6 total outcomes on a die, the probability of rolling an even number on one roll is \( \frac{3}{6} = \frac{1}{2} \).

Step 3: Determine independent event probability

Each die roll is independent. Therefore, to find the probability of rolling an even number three times, you multiply the probabilities of rolling an even number on each individual roll.

Step 4: Calculate probability for three rolls

Using the probability from Step 2, the probability of rolling an even number three times is \( \left( \frac{1}{2} \right)^3 \).

Step 5: Simplify the result

Simplify \( \left( \frac{1}{2} \right)^3 \) to get the final probability: \( \frac{1}{8} \).

Key Concepts

Independent Events

In probability, independent events are those whose outcomes do not influence one another. This means each event has its own separate outcome, and knowing the result of one does not change the probability of the other occurring. When you roll a die, each roll is considered an independent event. What you roll first has no impact on what result you might get on the next roll. This independence is crucial when calculating probabilities for multiple rolls or attempts. By treating them as separate occurrences, it allows us to multiply probabilities, a key concept in finding combined probabilities of sequences of independent events.

Die Roll Probability

A standard six-sided die has an equal chance of landing on any one of its sides, numbered 1 through 6. Because all sides are equally probable, each number has a probability of appearing which is calculated by dividing 1 by the total number of sides.This gives us:

• Probability of any one number: \( \frac{1}{6} \)

Knowing this basic principle helps us determine probabilities of more specific outcomes, such as rolling an even number or forming a sequence of numbers over multiple rolls. When you want to find the probability of rolling something specific, you count how many ways it can happen out of the total options, making it easy to calculate scenarios like finding an even number or rolling doubles.

Even Numbers

Even numbers on a standard die are those which are divisible by 2 without a remainder. In a typical situation with rolling dice, the even numbers are 2, 4, and 6.This number setup means there are 3 out of 6 possible outcomes in this category:

• Even numbers: 2, 4, 6
• Probability of rolling an even number: \( \frac{3}{6} = \frac{1}{2} \)

Understanding even numbers is essential when determining the probability of rolling multiple even numbers in succession, as it sets the foundation for calculations involving sequences and repetition in events.

Probability Calculation

Calculating probabilities involves understanding ratios and likelihoods. For multiple independent events, such as rolling a die several times, the calculation hinges on multiplying individual event probabilities together.Here is a step-by-step breakdown:

• Identify probability of a single event: For rolling an even number, it's \( \frac{1}{2} \)
• Calculate for multiple events: For three consecutive even rolls, use \( \left( \frac{1}{2} \right)^3 \)
• Simplify the result: \( \left( \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \right) = \frac{1}{8} \)

The value \( \frac{1}{8} \) represents how likely it is to achieve an even number on each roll over three attempts. This process highlights the importance of understanding independent probabilities and multiplication to reach accurate outcomes in probability scenarios.