Problem 18 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A pair of fair dice is rolled. What is the probability that the number landing uppermost on the first die is a 4 if it is known that the sum of the numbers landing uppermost is \(7 ?\)

Step-by-Step Solution

Answer

The probability that the number landing uppermost on the first die is a 4, given that the sum of the numbers landing uppermost is 7, is \(\frac{1}{6}\).

Step 1: Determine the number of favorable outcomes

We need to find the total number of outcomes where the first die shows 4, and the sum of the two dice is 7. There is only one such outcome, when the first die is a 4 and the second die is a 3 (4+3=7).

Step 2: Determine the number of possible outcomes for the given condition

To find the number of possible outcomes where the sum is 7, consider all the pairs of numbers on two dice that add up to 7. There are the following possibilities: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1). There are 6 possible outcomes where the sum of the two dice is 7.

Step 3: Calculate the probability

We can calculate the probability by dividing the number of favorable outcomes by the number of possible outcomes for the given condition, which is: \[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}} = \frac{1}{6} \] So, the probability that the number landing uppermost on the first die is a 4, given that the sum of the numbers landing uppermost is 7, is \(\frac{1}{6}\).

Key Concepts

Conditional probability

Conditional probability is a fundamental concept in probability theory. It is the probability of an event occurring, given that another event has already occurred. In simple terms, it answers the likelihood of a specific event under certain conditions. For example, in our dice game scenario, we want to determine the probability that the first die shows a 4, given that we know the total sum of the dice is 7.To calculate this, we use the formula for conditional probability:\[P(A | B) = \frac{P(A \cap B)}{P(B)}\]Where:

• \(P(A | B)\): The conditional probability of event A given event B.
• \(P(A \cap B)\): The probability of both events A and B happening together.
• \(P(B)\): The probability of event B.

By using this formula, we can understand the relation between two interdependent events. It helps in scenarios where the outcome of one event influences the probability of another. This concept is widely employed in various fields, such as statistics, finance, and risk assessment.

Combinatorics

Combinatorics is a branch of mathematics dealing with the counting, arrangement, and combination of objects. When discussing dice games, combinatorics is crucial. It allows us to systematically count possibilities, facilitating the calculation of probabilities. In our example, we determined all possible dice combinations. They added up to the given condition, which was a sum of 7. The possible combinations were (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). Combinatorics enabled us to verify there are exactly six possible outcomes. Understanding these combinations is essential for calculating probabilities. By effectively enumerating each possibility, we ensure no valid outcome is missed. This avoids inaccuracies when calculating probabilities. In more complex scenarios, more advanced combinatorial methods might be necessary.

Dice games

Dice games rely heavily on randomness, where each roll produces an independent result. A fair die gives the player an equal chance for each number to land face up. Two dice rolls produce a wide range of possible combinations, particularly used to demonstrate probability concepts. In this example, each die can land on one of 6 faces. Rolling two dice simultaneously presents a total of 36 possible outcomes. By understanding this, players or observers can assess likelihoods of specific events. Dice games often serve as relatable examples for learning probability and statistics. They force us to think about outcomes and frequencies. As demonstrated with our example, we see how conditions (like a specific sum) influence potential results.