Problem 10 - Calculus for Biology and Medicine

Question

10\. You roll two fair dice. Find the probability that the first die is a 5 given that the minimum of the two numbers is a 3 .

Step-by-Step Solution

Answer

The probability is \( \frac{1}{4} \).

Step 1: Understand the Problem

We need to find the probability that the first die shows a 5, given that the minimum of the two dice is at least a 3.

Step 2: Identify Possible Outcomes

Since the minimum of the two numbers is a 3, the possible outcomes for the two dice are: (3,3), (3,4), (3,5), (3,6), (4,3), (5,3), (6,3), (4,4), (4,5), (4,6), (5,4), (5,5), (5,6), (6,4), (6,5), and (6,6). This gives us a total of 16 outcomes.

Step 3: Identify Favorable Outcomes

For the first die to be a 5, the outcomes must be (5,3), (5,4), (5,5), or (5,6). Therefore, there are 4 favorable outcomes.

Step 4: Calculate the Probability

The probability is the ratio of the number of favorable outcomes to the total number of possible outcomes. So, the probability is \( \frac{4}{16} = \frac{1}{4} \).

Key Concepts

Understanding Probability Theory

Probability theory is the branch of mathematics that deals with the analysis of random phenomena. Essentially, it provides us with the tools to quantify uncertainty and predict the likelihood of various possible outcomes. When we talk about probability, we are often referring to the chance that a particular event will occur. The basic formula for probability is:\[ P(A) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \]This relationship helps us in scenarios like rolling dice, drawing cards, or performing any random experiment. In the context of dice rolling, determining the probability involves counting up all possible outcomes and the specific ones that match the condition we're interested in. Probability theory helps us structure our thinking so we can make logical and statistically sound decisions, rather than relying on intuition alone. This allows us to calculate with precision the odds of specific combinations in games of chance.

Exploring Rolling Dice

Rolling dice is a classic example used in probability theory because it's both simple and instructive. A standard die has six faces, numbered from 1 to 6, and each face has an equal chance of landing face up.When two dice are rolled, which is often the case in games and probability exercises, the possible combinations multiply. With two dice, each die can land on any of the six numbers, so there are a total of \(6 \times 6 = 36\) possible outcomes.However, specific conditions can limit these outcomes. For example, if we are told that the minimum of two dice must be 3, only the combinations where at least one die shows a 3, or higher, are considered. That changes the total outcomes to a smaller, more manageable number, making it easier to calculate probabilities for other specific results within those constraints.

Applying Mathematical Reasoning

Mathematical reasoning is crucial for solving probability problems like the dice example provided. It involves logical thinking and the application of mathematical principles to deduce clear, accurate outcomes. In the context of this problem, we used reasoning to:

• Filter out non-relevant outcomes, such as those where neither die shows a minimum of 3.
• Focus on identifying which outcomes meet the condition of having the first die show a 5.
• Break down the problem into smaller, more manageable parts, such as calculating favorable versus total outcomes.

Once we have this clarity, we use mathematical reasoning to divide the number of favorable outcomes (when the first die shows a 5) by the total possible outcomes that satisfy the problem's condition. By applying logical steps one by one, we reach the clear conclusion that the probability of our particular interest is \( \frac{1}{4} \). Through practice and familiarity with these concepts, mathematical reasoning helps streamline our thinking for solving problems efficiently and accurately.