Problem 50 - College Algebra

Question

A single die is rolled twice. Find the probability of rolling an odd number the first time and a number less than 3 the second time.

Step-by-Step Solution

Answer

The probability of rolling an odd number the first time and a number less than 3 the second time is \(\frac{1}{6}\).

Step 1: Identify Probability of First Roll

The first roll requires an odd number. A standard die has six faces and the three odd numbers among them are 1, 3, and 5. So, the probability of getting an odd number on a single roll is defined as the desired outcomes divided by the total outcomes. In this case, it is 3 (odd numbers 1,3,5) out of 6 (total possible outcomes), simplified to \(\frac{1}{2}\).

Step 2: Identify Probability of Second Roll

The second roll requires a number less than 3. The numbers less than 3 on a die are 1 and 2. So the probability of rolling a number less than 3 on a single roll is 2 (numbers 1 and 2) out of 6 (total possible outcomes), simplified to \(\frac{1}{3}\).

Step 3: Combine The Probabilities

Since these two events are independent (each roll of the die does not affect any other roll), we can find the overall probability by multiplying these two probabilities together. So, \(\frac{1}{2}\) times \(\frac{1}{3}\) equals \(\frac{1}{6}\).

Key Concepts

Independent Events Probability

When we talk about independent events in probability, we are referring to scenarios where the outcome of one event does not influence the outcome of another event. This is a crucial concept because it allows us to calculate the probability of multiple events occurring by simply multiplying their individual probabilities.

For example, rolling a die is often used to illustrate independent events. Each roll is separate from the previous, meaning the result of one roll does not affect the next. When determining the combined probability of two independent events occurring, like rolling a die twice and getting an odd number the first time and a number less than 3 the second time, you multiply the two probabilities: \(\frac{1}{2} \text{ for odd number}\) times \(\frac{1}{3} \text{ for less than 3}\) equals \(\frac{1}{6}\).

This principle applies to a wide range of scenarios, not just dice, such as drawing cards from a deck (if replaced after each draw), flipping coins, or even selecting random samples from a population.

Single Die Roll Probability

The probability of a single die roll resulting in a specific outcome can often be understood as a simple fraction, where the numerator is the number of favorable outcomes and the denominator is the number of possible outcomes. A standard six-faced die provides a perfect context for illustrating this principle.

For instance, when aiming to roll an odd number, we know that a die has three odd numbers—1, 3, and 5. Therefore, the probability of rolling an odd number is \(\frac{3}{6}\), which simplifies to \(\frac{1}{2}\). Similarly, rolling a number less than 3 gives us two favorable outcomes—1 and 2—out of the six faces, so the probability for this event is \(\frac{2}{6}\) or \(\frac{1}{3}\).

Understanding this concept helps in analyzing the likelihood of events and establishing expectations based on mathematical reasoning rather than intuition or guesswork.

Probability Concepts

Probability is often perceived as a measure of how likely an event is to occur, expressed on a scale from 0 to 1, where 0 indicates impossibility and 1 denotes certainty. Basic probability concepts include events, outcomes, favorable outcomes, and the probability formula.

The formula for probability is: \(P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\) where \(P(E)\) represents the probability of event \(E\) occurring.

Several additional principles are fundamental in understanding probability:

• Sample Space: The set of all possible outcomes.
• Event: A subset of the sample space.
• Favorable Outcomes: The outcomes that define the occurrence of the event.
• Addition Rule: Used to calculate the probability of one of several mutually exclusive events occurring.
• Multiplication Rule: Used to calculate the probability of two independent events both occurring, by multiplying their individual probabilities.

These concepts form the backbone for answering a myriad of probabilistic questions and developing a systematic approach to analyzing random phenomena in various disciplines, from simple games of chance to complex scientific studies.