Problem 26 - Calculus for Biology and Medicine

Question

Roll two fair dice and find the probability that the sum of the two numbers is even.

Step-by-Step Solution

Answer

The probability that the sum is even is \( \frac{1}{2} \).

Step 1: Understand the Total Outcomes

When rolling two dice, each die has 6 faces. Therefore, the total number of possible outcomes when rolling two dice is calculated by multiplying the number of faces on the first die by the number of faces on the second die: \[6 \times 6 = 36\] Thus, there are 36 possible outcomes when rolling two dice.

Step 2: Identify Favorable Outcomes

For the sum of two numbers to be even, it can either be (even + even) or (odd + odd). Each die can roll an even number (2, 4, 6) or an odd number (1, 3, 5). - Combination of even numbers: (2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4), (6, 6) - Combination of odd numbers: (1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5, 5) Therefore, there are 18 favorable outcomes.

Step 3: Calculate Probability

Probability is calculated as the number of favorable outcomes divided by the total number of possible outcomes. Thus, the probability that the sum of the two numbers is even is:\[ P(\text{even sum}) = \frac{18}{36} = \frac{1}{2} \]This simplifies to a probability of \( \frac{1}{2} \), indicating it's equally likely for the sum to be even or odd.

Key Concepts

Even and Odd Numbers

Understanding even and odd numbers is key in solving probability exercises involving dice. An even number is an integer divisible by 2 without a remainder, such as 2, 4, or 6. An odd number, on the other hand, leaves a remainder of 1 when divided by 2, examples being 1, 3, or 5. In dice rolls, results are integers ranging from 1 to 6. This splits neatly into three even numbers: 2, 4, and 6, and three odd numbers: 1, 3, and 5. This distribution allows us to explore certain patterns and interactions when evaluating the outcomes of rolling dice, specifically whether the sum of two dice rolls results in an even or odd number. Understanding these differences forms a foundation that guides us to determine probabilities effectively.

Dice Outcomes

When rolling dice, each face represents a different outcome. A standard die has six faces, each showing a different number: 1, 2, 3, 4, 5, or 6. Thus, a single roll of one die can result in any of these six numbers, making the outcomes quite manageable. When two dice are rolled together, the possible outcomes increase exponentially. Each die has 6 potential outcomes, creating a total of 36 outcomes (6 from the first die multiplied by 6 from the second die), such as (1,1), (1,2), etc.

These 36 different combinations need to be accounted for when determining the probability of certain conditions, like forming an even sum. Understanding these outcomes is crucial for assessing any probability as it establishes the total set of possibilities that dictate the denominator in probability calculations.

Favorable Outcomes

In probability, 'favorable outcomes' refer to the specific conditions that meet the criteria we're interested in. For our exercise, a favorable outcome is an event where the sum of numbers from two dice is even. This occurs through specific combinations of dice results.

To achieve an even sum, you can either add two even numbers or two odd numbers. For example, a roll resulting in (2 + 4) or (3 + 5) would be favorable, as both situations yield an even sum. In the example given, we identified 18 favorable outcomes in total through the possible combinations of (2, 4, 6) for even results and (1, 3, 5) for odd results on both dice. Understanding this concept enables students to pinpoint exactly which outcomes play into the calculation of probabilities in their specific context.

Total Possible Outcomes

The term 'total possible outcomes' is fundamental in understanding probability. It signifies every possible outcome that can occur from a randomness-inducing event, like rolling dice. For two dice, total possible outcomes are found by multiplying 6 outcomes per die, summing up to a comprehensive 36 outcomes.

This full set of outcomes acts as a base number for calculating probabilities. For any given question about dice, knowing the total number of outcomes helps determine the denominator in a probability fraction. In our scenario, any subset of outcomes (such as the 18 combinations for an even sum) is compared to this total number of 36 to form a probability ratio. This basic arithmetic forms the cornerstone of calculating probabilities and should be fully comprehended to tackle similar probability questions effectively.