Problem 26 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A pair of fair dice is rolled. What is the probability that a. The sum of the numbers shown uppermost is less than 5 ? b. At least one 6 is rolled?

Step-by-Step Solution

Answer

a. The probability that the sum of numbers shown uppermost is less than 5 is \( \frac{1}{6} \). b. The probability that at least one 6 is rolled is \( \frac{11}{36} \).

Step 1: Calculating the probability that the sum of numbers is less than 5

To solve this problem, first, identify all combinations that would result in a sum less than 5. The possible sums are 2, 3 and 4. - There is one scenario with a sum of 2, which is when both dice show 1. - For a sum of 3, there are two scenarios: 1 and 2, 2 and 1. - For a sum of 4, there are three scenarios: 1 and 3, 3 and 1, 2 and 2. Therefore, there are six favorable outcomes. Hence, the probability is \( \frac{6}{36} = \frac{1}{6}\).

Step 2: Calculating the probability of at least one 6 being rolled

All possible outcomes where at least one die shows 6 should be considered. - There are six scenarios where the first die shows 6 and the second die shows 1 to 6. - Similarly, there are six scenarios where the second die shows 6 and the first die shows 1 to 6. - But since both dice showing 6 is included in both the above cases, we should subtract it so that we only count it once to avoid duplication. Therefore, there are 11 favorable outcomes (6 + 6 - 1). Hence, the probability is \( \frac{11}{36} \). So, the answer to the problems are: a. The probability that the sum of numbers shown uppermost is less than 5 is \( \frac{1}{6} \). b. The probability that at least one 6 is rolled is \( \frac{11}{36} \).

Key Concepts

Dice Probability

When we talk about dice probability, we focus on understanding the likelihood of various outcomes when rolling dice. A standard die has six faces, each marked with a number from 1 to 6. When rolling two dice, the total number of outcomes becomes 36, as each die provides six options, and these are independent events.
The principle of dice probability involves counting the number of ways an event can happen and dividing it by the total number of possible outcomes. For example, if you want to know the probability of rolling a sum less than 5 with two dice, you identify all combinations that produce sums of 2, 3, or 4. Each combination aligns with a specific pair of numbers, such as rolling a 1 on the first die and a 3 on the second.
Understanding dice probability forms the foundation for many basic probability questions, letting you predict outcomes from simple games like rolling dice to more complex scenarios like strategic board games.

Combinatorics

Combinatorics is a branch of mathematics focusing on counting, arranging, and finding patterns in sets of objects. When rolling two dice, combinatorics helps us determine the number of ways specific outcomes can happen.
For instance, to find combinations leading to a sum of less than 5 on two dice:

• A sum of 2 occurs in only one way: rolling 1 on both dice.
• A sum of 3 can happen in two specific ways: rolling a 1 and 2, or 2 and 1.
• A sum of 4 appears in three different ways: 1 and 3, 3 and 1, or 2 and 2.

Adding these combinations, we find there are six favorable combinations when aiming for a sum less than 5. Combinatorics doesn't just make counting easier; it lays the groundwork for solving many probability problems by simplifying what seems complex.

Probability Calculation

Probability calculation is a crucial step in determining how likely an event is to occur. This involves identifying the ratio of favorable outcomes to the total possible outcomes of an event.
For example, calculating the probability of rolling at least one 6 with two dice focuses on finding how many rolls of the dice meet this condition. First, consider all scenarios where the first die is a 6 (six possibilities). Then, consider scenarios where the second die is a 6, also six cases.
Since rolling two sixes overlaps both groups, one overlap must be removed from the count to avoid duplicating this outcome. Thus, there are 11 favorable outcomes. The probability formula, \[ P = \frac{\text{Number of Favorable Outcomes}}{\text{Total Possible Outcomes}} \]becomes\[ P = \frac{11}{36}. \]Mastering probability calculations helps in making informed predictions and understanding events across various fields, from gaming strategies to statistical data analysis.