Problem 23 - College Algebra with Modeling and Visualization

Question

Find the probability of the compound event. Rolling a die three times and obtaining a 5 or 6 on each roll

Step-by-Step Solution

Answer

The probability is \( \frac{1}{27} \).

Step 1: Understand the Event

The problem asks for the probability of rolling a 5 or a 6 on a die three times in a row. This event consists of three independent occurrences (each die roll), which all need to be 5 or 6.

Step 2: Calculate the Probability for a Single Roll

For a single die roll, the numbers 5 and 6 are two out of the six possible outcomes. Therefore, the probability of rolling either a 5 or a 6 on one roll is \( \frac{2}{6} = \frac{1}{3} \).

Step 3: Use the Rule for Independent Events

Since the rolls are independent, the probability of all three events happening (rolling a 5 or 6 in each roll) is the product of the probabilities of the individual events occurring. That is, \( \left( \frac{1}{3} \right)^3 \).

Step 4: Compute the Probability

Compute the probability of rolling a 5 or 6 in all three rolls: \( \left( \frac{1}{3} \right) \times \left( \frac{1}{3} \right) \times \left( \frac{1}{3} \right) = \frac{1}{27} \).

Key Concepts

Compound Event

In probability, a compound event refers to an event that involves the combination of two or more individual events. When considering a compound event, we aim to find the probability that two or more simple events will occur together. In the context of this exercise, rolling a die three times to get a 5 or a 6 each time is a compound event.
Compound events can be broken down into simpler parts, which are the occurrences of each event happening individually. To successfully calculate the probability of a compound event, it's essential to understand whether these individual events affect each other, or if they are independent.

Independent Events

Independent events are events where the outcome of one event does not influence the outcome of another. In simple terms, knowing the result of one event doesn't change the probability of another. Using the die example, rolling a die multiple times are independent events because the result of one roll does not affect the result of another.
Understanding this concept is crucial in solving the exercise because each die roll is a separate event, and the outcome of any roll does not alter the likelihood of the others. This affects how we approach probability calculation for compound events, by allowing us to multiply the probabilities of individual events to find the compound probability.

Probability Calculation

Probability is a way of quantifying how likely an event is to occur. When dealing with independent events, like rolling a die three times, calculating the probability involves some straightforward concepts.
To find the probability of the compound event that each die roll results in a 5 or a 6, first determine the probability for a single roll. The sides 5 and 6 represent two favorable outcomes out of six possible outcomes, so the probability is \( \frac{2}{6} = \frac{1}{3} \).
For independent events, the probability of a compound event occurring is the product of the probabilities of each individual event. Mathematically, this is represented as \( \left( \frac{1}{3} \right)^3 \) for this problem. Finally, compute \( \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27} \). This gives us the probability of rolling either a 5 or 6 three times in a row.