Problem 47 - College Algebra

Question

A single die is rolled twice. Find the probability of rolling a 2 the first time and a 3 the second time.

Step-by-Step Solution

Answer

The probability to roll a 2 the first time and a 3 the second time is 1/36.

Step 1: Calculate the probability of rolling a 2

For a single roll of a six-sided fair die (i.e., all outcomes {1,2,3,4,5,6} are equally likely), the probability to get a 2 is 1/6.

Step 2: Calculate the probability of rolling a 3

Similarly, the probability to get a 3 in a single roll of a six-sided die is also 1/6.

Step 3: Combine the probabilities

Since these are independent events, the overall probability of both happening is the product of their individual probabilities: \(1/6 \times 1/6 = 1/36.\)

Key Concepts

Probability Theory

At its core, probability theory is a mathematical framework for quantifying the likelihood of different outcomes in an uncertain situation. It is used to predict the chance of events happening. For instance, it can tell us the likelihood of rolling a specific number on a die or drawing a particular card from a deck.

To calculate the probability of a single event, we use the formula: \[\begin{equation} P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \end{equation}\]In the example of rolling a die, with six faces, the total number of possible outcomes is six. So, if we want to find out the probability of rolling a 2, we identify the number of favorable outcomes (in this case, rolling a 2) which is 1, and divide it by the total outcomes: \[\begin{equation} P(\text{rolling a 2}) = \frac{1}{6} \end{equation}\]

Understanding this fundamental framework of probability is crucial for solving a wide range of problems from gaming strategies to real-world applications like risk assessment and predictive modeling.

Independent Events in Probability

In probability theory, events are said to be independent if the occurrence of one event does not affect the probability of the other occurring. In other words, the outcome of one event has no influence on the outcome of another.

For example, when we roll a die, the result of the first roll does not affect the result of the second roll. Each roll is a separate event with its own set probability. When calculating the overall probability of independent events happening together, we multiply their individual probabilities.

Continuing with our dice roll exercise, since the occurrence of a 2 on the first roll and a 3 on the second roll are independent events, their combined probability is: \[\begin{equation} P(\text{first roll}=2 \text{ and } \text{second roll}=3) = P(\text{first roll}=2) \times P(\text{second roll}=3) \end{equation}\]Hence, the probability of rolling a 2 first and a 3 second on a six-sided die would be: \[\begin{equation} \frac{1}{6} \times \frac{1}{6} = \frac{1}{36} \end{equation}\]

Knowing whether events are independent or not is vital for accurate probability calculations.

Combinatorial Probability

Combinatorial probability deals with the likelihood of events occurring where there are several combinations or arrangements possible. It is an area within probability theory that combines the study of combinations and permutations. This is especially useful when we need to consider all the different ways in which events can happen.

For simple cases, like rolling a six-sided die twice, the combinatorial aspects are straightforward since the die outcomes for each roll don't depend on previous rolls. So, we multiply the probability of the individual outcomes to get the combined probability. However, in more complex situations with multiple objects, like selecting cards from a hand or arranging people in a line, using combinatorial mathematics becomes essential to count the number of favorable outcomes accurately.

For example, if we wanted to know the probability of drawing two specific cards from a deck in a row, we would need to account for the changing number of cards after the first draw. This complexity is where combinatorial probability shines, ensuring we consider the required combinations to calculate the chances accurately.