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: Identify Total Possible Outcomes

When rolling two dice, each die has 6 possible outcomes from 1 to 6. Thus, the total number of outcomes for two dice is calculated as follows: \[ 6 \times 6 = 36 \]

Step 2: Determine Favorable Outcomes for Even Sum

The sum of the numbers on two dice is even if and only if either both numbers are even or both numbers are odd. - Even outcomes (both even): (2,2), (2,4), (2,6), (4,2), (4,4), (4,6), (6,2), (6,4), (6,6) - 9 outcomes - Odd outcomes (both odd): (1,1), (1,3), (1,5), (3,1), (3,3), (3,5), (5,1), (5,3), (5,5) - 9 outcomes Total favorable outcomes = 9 + 9 = 18

Step 3: Calculate Probability

The probability of an event is the ratio of favorable outcomes to the total number of possible outcomes. Using results from steps 1 and 2, the probability that the sum is even is: \[ \frac{18}{36} = \frac{1}{2} \]

Step 4: Simplify Result and Conclusion

The probability simplifies to \( \frac{1}{2} \), which means there is a 50% chance that the sum of the numbers on two dice will be even.

Key Concepts

Dice

In probability exercises, dice are commonly used tools to understand random outcomes and how they relate to probabilities. A standard die is a cube with six faces, each displaying a number of dots ranging from 1 to 6. Every face is equally likely to land facing upwards, meaning each face has a probability of \(\frac{1}{6}\) when a single die is rolled.

When two dice are rolled, the scenario becomes slightly more complex, as every face of one die can pair with every face of the other die. This multiplies the number of possible results, leading to combinations that can be calculated by multiplying the number of sides on one die by the number on the other:

• 1 die: 6 outcomes
• 2 dice: \(6 \times 6 = 36\) outcomes

Understanding this principle is critical when tackling exercises that involve multiple dice, as it forms the basis for calculating further outcome probabilities.

Even Sum

An even sum requires that the total produced by rolling two dice is an even number. To arrive at an even result, either both dice must display an even number or both must show an odd number.

Here is a breakdown:

• Even Even Pair: (2,2), (2,4), (2,6), (4,2), (4,4), (4,6), (6,2), (6,4), (6,6)
• Odd Odd Pair: (1,1), (1,3), (1,5), (3,1), (3,3), (3,5), (5,1), (5,3), (5,5)

The total number of ways to achieve an even sum is computed by counting these pairs: 9 ways to achieve from uneven dice, 9 ways from odd dice, equaling 18 ways in total.

Recognizing these patterns—and more importantly, understanding why they yield the same results—facilitates the calculation of the probability of rolling an even sum.

Outcomes

Outcomes in probability refer to the possible results that can occur. In the context of rolling two dice, each individual outcome is a combination of numbers displayed on each die. There are 36 possible outcomes, derived from pairing each of the 6 sides of one die with each of the 6 sides of the other.

These outcomes are foundational to calculating probability:

• Total number of outcomes when rolling two dice: 36
• Favorable outcomes for obtaining an even sum: 18

Once you list all possible outcomes and identify the favorable ones, the concept of probability becomes straightforward. The probability is simply the portion of these favorable events relative to the total number of outcomes. Hence, the probability that the sum is even is \[\frac{18}{36} = \frac{1}{2}\] or 50% chance. This illustrates how the systematic counting of outcomes enables understanding of the likelihood of various events.