Problem 14 - Thinking Mathematically

Question

In Exercises 11-14, 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: Determine the probability of each individual event

The probability of rolling an odd number on a six-sided die is \( P(Odd) = \frac{3}{6} = \frac{1}{2} \) since there are 3 favorable outcomes (1, 3, or 5) out of 6 possible outcomes. The probability of rolling a number less than 3 (1 or 2) is \( P(<3) = \frac{2}{6} = \frac{1}{3} \) since there are 2 favorable outcomes out of 6 possible outcomes.

Step 2: Use the multiplication rule

Given that these events are independent (the result of the first roll does not affect the result of the second), the multiplication rule of probability allows for multiplying the individual probabilities together to find the overall probability. Therefore, \( P(Odd and <3) = P(Odd) * P(<3) \).

Step 3: Calculate the overall probability

Substitute Step 1 results into the equation from Step 2 to get the answer \( P(Odd and <3) = \frac{1}{2} * \frac{1}{3} = \frac{1}{6} \).

Key Concepts

Independent Events

When it comes to understanding probability, the concept of independent events is essential. Independent events are two or more events where the outcome of one event does not influence the outcome of another. For example, rolling a die twice involves two independent events, because how the die lands on the first roll doesn't affect how it will land on the second roll.

In the exercise provided, when a single die is rolled twice, the first roll and the second roll are independent events. Think of each roll as a fresh start, where the die has no memory of the previous result. Because the events do not affect each other, we can use specific rules to calculate the combined probability of independent events happening in sequence, which brings us to the next section—the multiplication rule of probability.

Multiplication Rule of Probability

The multiplication rule of probability is a fundamental concept that is used when dealing with independent events. Simply put, it states that if two events, A and B, are independent, then the probability of both events occurring is equal to the product of their individual probabilities: \[ P(A \text{ and } B) = P(A) \times P(B) \].

Following our example, after establishing that rolling an odd number and rolling a number less than 3 are independent events, we can multiply their individual probabilities to find the probability of both happening. Taking the calculated probabilities from the exercise, \( P(Odd) = \frac{1}{2} \) and \( P(<3) = \frac{1}{3} \), we apply the multiplication rule: \( P(Odd \text{ and } <3) = P(Odd) \times P(<3) = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} \). Through this rule, probability problems involving independent events are straightforward to solve by simply using multiplication.

Favorable Outcomes

In probability, favorable outcomes refer to the specific results of an event that satisfy the condition we are interested in. The probability of an event occurring is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. It's crucial to correctly identify which outcomes are favorable to solve the problem accurately.

For a die roll, there are six possible outcomes (1 through 6). To solve for the probability of rolling an odd number first, you identified the favorable outcomes as 1, 3, and 5. Therefore, with 3 favorable outcomes out of 6, the probability is \( \frac{3}{6} = \frac{1}{2} \). Then, for rolling a number less than 3, the only favorable outcomes are 1 and 2, making the probability \( \frac{2}{6} = \frac{1}{3} \). This step is key to establishing the foundation for applying the multiplication rule of probability for independent events.