Problem 49 - College Algebra

Question

A single die is rolled twice. Find the probability of rolling an even number the first time and a number greater than 2 the second time.

Step-by-Step Solution

Answer

Therefore, the probability of rolling an even number the first time and a number greater than 2 the second time is \(\frac{1}{3}\).

Step 1: Determine the Probability of Rolling an Even Number

An even number on a die is 2, 4 or 6. Given that a die has 6 faces, the probability of rolling an even number is the number of even outcomes (which is 3) divided by the total number of outcomes (which is 6). Therefore, the probability is \(\frac{3}{6}\) or \(\frac{1}{2}\).

Step 2: Determine the Probability of Rolling a Number Greater than 2

The numbers on a die that are greater than 2 are 3, 4, 5 and 6. Given that a die has 6 faces, the probability of rolling a number greater than 2 is the number of suitable outcomes (which is 4) divided by the total number of outcomes (which is 6). Therefore, the probability is \(\frac{4}{6}\) or \(\frac{2}{3}\).

Step 3: Multiply Probabilities of Independent Events

Given that the two events are independent (the outcome of the first roll does not impact the outcome of the second roll), the total probability of both events occurring is the product of the two probabilities. This will be \(\frac{1}{2}\) (probability of rolling an even number) times \(\frac{2}{3}\) (probability of rolling a number greater than 2), which equals \(\frac{1}{3}\).

Key Concepts

Independent Events Probability

When studying probability, one core concept to understand is the idea of independent events. These are events where the outcome of one event does not affect or change the outcome of another. In simpler terms, whatever happens in the first event has no bearing on what will happen next.

For example, when you roll a die, whatever number comes up on the first roll, it doesn't influence what will come up on the second roll. They are independent. To calculate the total probability of independent events occurring together, you multiply the probability of each event. For instance, if the probability of event A is \( p(A) \) and the probability of event B is \( p(B) \) and A and B are independent, then the probability of both events happening, \( p(A \text{ and } B) \), is \( p(A) \times p(B) \).

Rolling Dice Probability

Rolling dice is a classic example used in probability exercises because it’s an easy-to-understand scenario with clear outcomes. A standard die has six faces, each with a number from 1 to 6. Therefore, when a single die is rolled, there are six possible outcomes.

Since each face of a fair die is equally likely to land face up, each roll is independent of the other, and the probability of rolling any single number is \( \frac{1}{6} \). Understanding these basics is crucial in solving more complex problems, like calculating the likelihood of rolling certain numbers across multiple dice rolls. It's also important to remember the rules surrounding independent events when considering multiple dice rolls: the outcome of the previous rolls does not influence the outcome of the future rolls.

Even Number Probability

In the context of rolling dice, calculating the probability of getting an even number is a straightforward task. The numbers that are considered even on a six-sided die are 2, 4, and 6. These are three out of the six possible outcomes, which means the probability of rolling an even number on one roll is \( \frac{3}{6} \), or simplified to \( \frac{1}{2} \).

Hence, each time you roll the die, you have a 50% chance, or a probability of 0.5, of landing on an even number. This concept underlines an essential part of introductory probability — determining the likelihood of a particular type of outcome (like an even number) when dealing with a discrete and finite set of possibilities (like the faces of a die).