Problem 1 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A pair of dice is rolled, and the number that appears uppermost on each die is observed. Refer to this experiment and find the probability of the given event. The sum of the numbers is an even number.

Step-by-Step Solution

Answer

The probability of getting an even sum when rolling a pair of dice is 1/2 or 50%. This is calculated by finding the total number of possible outcomes (36) and dividing the number of favourable outcomes (18) by the total possible outcomes.

Step 1: Identify the total number of possible outcomes

When rolling a pair of dice, there are 6 sides on each die, so there are 6 * 6 = 36 possible outcomes.

Step 2: Identify the favourable outcomes (even sums)

A sum is even if the numbers being added are both even or both odd. We can list the possible combinations as follows: Even sums (both dice show even numbers): (2, 2), (2, 4), (2, 6) (4, 2), (4, 4), (4, 6) (6, 2), (6, 4), (6, 6) Even sums (both dice show odd numbers): (1, 1), (1, 3), (1, 5) (3, 1), (3, 3), (3, 5) (5, 1), (5, 3), (5, 5) There are 9 even sum outcomes for both even and odd combinations, which gives us a total of 9 + 9 = 18 favourable outcomes.

Step 3: Calculate the probability

To find the probability of getting an even sum, we need to divide the number of favourable outcomes (even sums) by the total number of possible outcomes: Probability = (Number of favourable outcomes) / (Total number of possible outcomes) Probability = 18 / 36 = 1/2 So, the probability of getting an even sum when rolling a pair of dice is 1/2 or 50%.

Key Concepts

Probability Theory

When we talk about probability theory, we're diving into a branch of mathematics that deals with quantifying how likely events are to happen. It's a way of making sense of randomness and can be applied to many real-world scenarios, such as rolling dice in our exercise.

In this dice-rolling experiment, probability theory helps us calculate the likelihood of rolling an even sum. To do this, we follow a basic principle: the probability of an event occurring equals the number of favorable outcomes divided by the total number of possible outcomes. In mathematical terms, this probability is represented by the formula \( P(E) = \frac{n(E)}{n(S)} \), where \( P(E) \) is the probability of event \( E \) occurring, \( n(E) \) represents the number of favorable outcomes, and \( n(S) \) denotes the total number of possible outcomes in sample space \( S \).

In our dice example, each die has 6 sides, therefore when rolling two dice, the sample space (the set of all possible outcomes) consists of 36 combinations. To find the probability of rolling an even total, we need to count all the pairs that add up to an even number. By identifying these as favorable outcomes, we can use the probability formula to calculate the odds of this event.

Combinatorics

Combinatorics is a field of mathematics concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It's the art of systematic counting without actually counting every single possibility, which is especially useful when dealing with larger sets.

In our dice problem, we use combinatorics to list and count the number of favorable outcomes. When it comes to figuring out how many ways we can roll a pair of dice to achieve an even total, combinatorics principles come into play. Here, we need to consider combinations of dice where the sum is even. For two dice, these combinations could involve either both dice showing even numbers or both dice showing odd numbers. We quickly find there are 18 such combinations.

Why is this efficient?

Imagine if instead of two dice, we had three or more. It would become increasingly unwieldy to list out every combination. Combinatorics gives us tools and formulas to handle these situations more efficiently, such as using permutations and combinations to calculate the number of favorable outcomes without exhaustive listing.

Mathematical Statistics

Mathematical statistics is a branch of mathematics that uses probability theory to analyze and interpret data. It provides us with the tools and methods to make inferences about a population, based on sample data. In the context of our dice experiment, mathematical statistics isn't used directly, since we know the exact probabilities from the combinatorial analysis. However, the concept is very relevant when we extend our understanding to more complex scenarios.

For instance, if we were to repeatedly roll a pair of dice and record our results to estimate the probability of rolling an even sum rather than calculating it theoretically, we would be venturing into the realm of mathematical statistics. Suppose our experiment runs many times, and each time we note whether the sum of the dice is even. We could then use statistical methods to estimate the probability of the even sum based on our experimental data.

Importance in Data Analysis

In situations where we can't calculate probabilities directly, like in real-world experiments or sample surveys, mathematical statistics is invaluable. It allows us to use sample data to make predictions or decisions regarding the larger population, which is a cornerstone of data analysis.