Problem 12 - Thinking Mathematically

Question

In Exercises 11-14, a single die is rolled twice. Find the probability of rolling a 5 the first time and a 1 the second time.

Step-by-Step Solution

Answer

The probability of rolling a 5 the first time and a 1 the second time is \( \frac{1}{36} \).

Step 1: Understand the problem

We are rolling a fair die twice and need to find the probability that the first roll results in 5 and the second roll, in 1. Since the rollings are independent, the joint probability of two events is equal to the product of their individual probabilities.

Step 2: Calculate the probabilities

The probability of rolling a 5 (or any specific number from 1 to 6) on a fair die is \( F = \frac{1}{6} \)since there is one '5' on the die and six possible outcomes in total. Similarly, the probability of rolling a 1 is \( S = \frac{1}{6} \)

Step 3: Calculate the joint probability

We multiply the individual probabilities to get the joint probability of both events happening: \( P = F * S = \frac{1}{6} * \frac{1}{6} = \frac{1}{36} .\)

Key Concepts

Independent Events

When studying probability, it's essential to understand the concept of independent events. Independent events are occurrences with outcomes that do not affect each other. In other words, the outcome of one event has no impact on the outcome of another. This can be compared with tossing a coin; whether you get heads or tails on one flip doesn't change the odds of getting heads or tails on the next flip.

To illustrate this with dice, let's imagine rolling a die two times in a row. The result of the first roll does not influence the result of the second roll. Each roll is completely separate and has the same probability of landing on any one of the six faces of the die. This means that the first and second rolls are independent events.

Joint Probability

Joint probability refers to the likelihood of two or more events happening at the same time. It's the 'and' probability, meaning we want to know the probability of event A and event B occurring together. To find the joint probability of two independent events, we multiply the probability of each event occurring separately.

So, if we want to calculate the joint probability of rolling a 5 on the first roll and a 1 on the second roll of a die, we multiply the individual probabilities of each event. If each outcome on the die has an equal chance of occurring, the probability of rolling a 5 is \( \frac{1}{6} \) and the probability of rolling a 1 is also \( \frac{1}{6} \). Therefore, the joint probability of rolling a 5 first and a 1 second is \( \frac{1}{6} * \frac{1}{6} = \frac{1}{36} \).

Rolling a Die

Rolling a die is a classic example used to explain basic probability because a standard die is a six-sided cube, with each side featuring a different number of dots, from 1 to 6. It's a perfect model for discussing uniform probability distributions; each outcome (rolling a 1, 2, 3, 4, 5, or 6) has an equal chance of occurring, which is \( \frac{1}{6} \), as there is only one of each number on the die.

In our specific exercise, we focus on rolling a die twice. Even though we perform two rolls, because of the principle of independence, the probability of landing on a particular number for each separate roll remains unchanged at \( \frac{1}{6} \). This simple tool is fundamental in understanding the basics of probability theory and calculating chances in more complex scenarios.