Problem 48 - College Algebra

Question

A single die is rolled twice. Find the probability of getting: 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 first and a number less than 3 second is 0.165.

Step 1: Calculate Probability of First Event

First recognize that getting an odd number when rolling a die means getting either 1, 3, or 5. The total possible outcomes are 6 (the numbers 1 through 6 on the die). So, the probability (P) of getting an odd number (A) is the number of successful outcomes divided by the total outcomes, i.e., P(A) = 3 / 6 = 0.5.

Step 2: Calculate Probability of Second Event

Getting a number less than 3 when rolling a die refers to getting either 1 or 2. With 6 total possible outcomes, the probability (P) of getting a number less than 3 (B) is P(B) = 2 / 6 = 0.33, rounded to two decimals.

Step 3: Calculate Combined Probability

Since the two events are independent, the probability of both events happening is the product of their individual probabilities. Therefore, the probability (P) of getting an odd number first (A) and getting a number less than 3 second (B) is P(A and B) = P(A) * P(B) = 0.5 * 0.33 = 0.165, rounded up to three decimals.

Key Concepts

Independent Events

The concept of independent events is critical to understanding the intricacies of probability, especially when dealing with multiple events such as dice rolls. Independent events in probability mean that the outcome of one event does not influence the outcome of another. For example, when a die is rolled twice, what you get on the first roll does not, in any way, affect what will come up on the second roll.

Dice rolls are a classic example used to illustrate independent events due to the die's lack of memory. Each roll is a separate and independent event. This is crucial when calculating the combined probability of two events that occur in sequence. When the events are independent, the probability of both occurring is the product of the individual probabilities of each event happening on its own.

Probability Calculation

Calculating probability is a fundamental task in mathematics and it involves determining the chances of a particular event happening. It is expressed as the ratio of the number of favorable outcomes to the total number of possible outcomes. In the context of dice, the probability of getting a certain result is calculated by dividing the number of ways that result can occur by the total number of possible outcomes (which, for a standard six-sided die, is six).

The initial step in probability calculation often involves a clear understanding of what qualifies as a 'successful' outcome. For instance, in our exercise, an odd number and a number less than 3 are considered successful outcomes for the respective rolls. The probability for each of these is calculated separately. Then, if the events are independent—as they are in the case of sequential die rolls—the probabilities are multiplied to find the combined probability of both events occurring.

Mathematical Probability

Mathematical probability is the numerical measure of the likelihood of an event, ranging from 0 (impossibility) to 1 (certainty). It is the basis of probability theory, which deals with the analysis of random events and is widely used in fields such as finance, insurance, and everyday decision making.

In mathematical probability, we rely on the principles of combinatorics and the laws of large numbers to predict the behavior of random events over time. If we take our dice roll example, by understanding the basic concept that there are six equally likely outcomes, we can determine the mathematical probability of any single roll. Extending this knowledge to sequences of rolls, or to more complex scenarios, builds on the foundation of knowing how to calculate individual event probabilities and combine them when events are independent.