Problem 11 - Thinking Mathematically

Question

In Exercises 11-14, 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 of rolling a 2 the first time and a 3 the second time is \(\frac{1}{36}\).

Step 1: Compute the Probability of Rolling a 2

The first step is to calculate the probability of rolling a 2 on a six-sided die. For a fair six-sided die, each face (and hence, each number 1 through 6) has an equal probability of being rolled, which is \(\frac{1}{6}\). Therefore, the probability of rolling a 2 is \(\frac{1}{6}\).

Step 2: Compute the Probability of Rolling a 3

Next, calculate the probability of rolling a 3 on a six-sided die. Just like the calculation in step one, each number has an equal probability of being rolled on a fair six-sided die, which means the probability of rolling a 3 is \(\frac{1}{6}\).

Step 3: Find the Combined Probability

The two events (rolling a 2 first and a 3 second) are independent of each other, which means the total probability of both events happening is the product of their individual probabilities. Thus, the combined probability is \(\frac{1}{6} * \(\frac{1}{6} = \(\frac{1}{36}\).

Key Concepts

Independent Events

When talking about probability, independence refers to two or more events that have no influence on each other's outcomes. In other words, the result of one event doesn't impact the likelihood of another. For instance, rolling a die twice involves independent events. The number that appears on the first roll does not affect or predict the number on the second roll. This is a key concept of independence in probability.

• Event independence means outcomes are separate.
• The result of one event doesn't change the probability of the other.
• With dice, each roll is unaffected by the previous one.

Understanding this concept helps us calculate the probability of multiple events occurring together because each is treated as if it happens in isolation from the others.

Combined Probability

To figure out the likelihood of two events happening in sequence, we use combined probability. For independent events, like rolling a die twice, combined probability is simply the product of the individual probabilities of each event.For example, if you want to find out the probability of rolling a 2 followed by a 3, you do the following:

• Probability of rolling a 2 = \(\frac{1}{6}\)
• Probability of rolling a 3 = \(\frac{1}{6}\)
• Combined probability = \(\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}\)

The combined probability tells us that the chance of these two events happening one after the other is 1 in 36. By multiplying the probabilities of independent events, we predict the occurrence of both, highlighting the principle of combined probability.

Fair Die

A fair die is an idealized six-sided die where each face has an equal chance of landing up when the die is rolled. This assumption of fairness is crucial in probability calculations because it ensures each outcome is equally likely. For a standard die, this means:

• Each face (numbers 1 through 6) has a probability of \(\frac{1}{6}\).
• No face is favored over another; the die is not biased or weighted.
• Rolling any specific number like 2 or 3 is equally probable.

Such a die provides a consistent environment for probability exercises, where each roll is an isolated event with a predictable probability. This consistency helps in understanding the theory of probability and in making accurate calculations for combinations of events.