Problem 36 - Discrete Mathematics with Applications

Question

A die is rolled four times. Find the probability of obtaining: At least one six.

Step-by-Step Solution

Answer

The probability of obtaining at least one six when rolling a die four times is \(\frac{671}{1296}\).

Step 1: Calculate the probability of not getting a six in a single roll.

There are six sides of a die, and only one of them is six. Thus, the probability of not getting a six in a single roll is: P(not rolling a six) = \(\frac{5}{6}\)

Step 2: Calculate the probability of not getting a six in four rolls.

Since the four rolls are independent events, we can multiply the probabilities of not rolling a six together. P(no sixes in four rolls) = \(\frac{5}{6}\) × \(\frac{5}{6}\) × \(\frac{5}{6}\) × \(\frac{5}{6}\) = \(\frac{5^4}{6^4}\)

Step 3: Calculate the probability of getting at least one six in four rolls.

Now, we will calculate the probability of getting at least one six in four rolls by subtracting the probability of not getting any sixes from 1. P(at least one six) = 1 - P(no sixes in four rolls) = 1 - \(\frac{5^4}{6^4}\)

Step 4: Simplify the probability.

Finally, we will simplify the probability by calculating the fraction. P(at least one six) = 1 - \(\frac{625}{1296}\) = \(\frac{671}{1296}\) So, the probability of obtaining at least one six when rolling a die four times is \(\frac{671}{1296}\).

Key Concepts

Discrete Mathematics

Discrete Mathematics is a branch of mathematics that deals with distinct and separate values. Unlike continuous systems, discrete mathematics focuses on countable, distinct structures.
This includes numbers that can assume only specific, separate values. An example is the roll of a die, which can result in any one of six distinct values: 1, 2, 3, 4, 5, or 6.

In probability, we often analyze discrete structures, such as dice rolls, coin tosses, or drawing cards from a deck. These discrete events help illustrate fundamental principles in statistics and probability.

• Each roll of a die is an independent event with a discrete number of possible outcomes.
• The probability of any specific outcome is fixed and countable.

Understanding these discrete processes is crucial in advanced fields like computer science and cryptography, as they allow for precise algorithms and computations.

Independent Events

In probability theory, independent events are crucial for understanding how multiple events interact. Two events are independent if the outcome of one does not affect the outcome of the other.
This is often symbolized as, if event A and event B are independent, then \[P(A \cap B) = P(A) \cdot P(B)\].

When rolling a die multiple times, each roll is treated as an independent event. This means that rolling a six on one occasion does not change the odds of rolling a six on the next.

• Each roll has a probability of \(\frac{1}{6}\) for a six since the die has six faces.
• The probability of not rolling a six is \(\frac{5}{6}\), and this probability stays the same across multiple rolls.

Understanding independence helps in calculating probabilities over multiple events without one influencing the other. This is key in solving problems with repeated trials, like dice rolls.

Complement Rule

The complement rule is a fundamental concept in probability that simplifies how we calculate certain events, especially when dealing with complex scenarios.
It states that the probability of an event happening is 1 minus the probability of it not happening: \[P(A) = 1 - P(A^c)\], where \(A^c\) is the complement of event A.

In simpler terms, finding the probability of an event not occurring helps us calculate the probability of the event's opposite occurring. In the die exercise, the task was to find the probability of getting at least one six in four rolls.

• First, we find the probability of not getting a six in four rolls.
• Then, apply the complement rule: 1 minus the probability of no sixes.

This technique is powerful, allowing us to solve problems effectively, particularly when it is easier to calculate the outcome of events not occurring than the outcomes themselves.