Problem 9 - Calculus for Biology and Medicine

Question

You roll two fair dice. Find the probability that the first die is a 4 given that the sum is 7 .

Step-by-Step Solution

Answer

The probability is \( \frac{1}{6} \).

Step 1: Understand the Total Outcomes When Sum is 7

First, calculate all possible outcomes when the sum of the numbers on two dice is 7. The possible outcomes are: \((1,6), (2,5), (3,4), (4,3), (5,2), (6,1)\). So, there are 6 favorable outcomes.

Step 2: Identify Favorable Outcomes for the Condition

Now identify how many of these outcomes have the first die showing 4. The relevant outcome from the list \((1,6), (2,5), (3,4), (4,3), (5,2), (6,1)\) is \((4,3)\). Hence, there is only 1 favorable outcome.

Step 3: Calculate the Conditional Probability

Use the conditional probability formula: \[ P( ext{First die is 4} \,|\, ext{Sum is 7}) = \frac{ ext{Number of favorable outcomes}}{ ext{Total outcomes when sum is 7}} = \frac{1}{6} \].

Key Concepts

Probability Theory

Probability theory is a branch of mathematics that deals with uncertainty and randomness. It provides a framework for quantifying the likelihood of events, and it’s widely used in various fields, from science to finance.
In probability theory, every outcome from a set of possible outcomes of a random experiment is assigned a probability value that lies between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
For experiments involving random events, such as rolling dice, probability theory helps us predict how often these events will occur over a long period. Understanding the basic terms:

• Experiment: An operation or process that leads to well-defined results called outcomes.
• Sample Space: The set of all possible outcomes (e.g., rolling a dice has a sample space of {1, 2, 3, 4, 5, 6}).
• Event: Any subset of a sample space (e.g., getting an even number when rolling a dice).
• Probability of an Event: The measure of the likelihood that an event will occur, calculated using relevant formulas specific to the problem at hand.

Dice Probability

Dice probability is a specific application of probability theory, related to the outcomes of rolling dice. When you roll a pair of fair dice, each die has 6 faces numbered 1 to 6, and each face is equally likely to appear.
Thus, every roll of the dice results in a pair of numbers, with 36 possible outcomes in total (since there are 6 choices for the first die and 6 for the second, 6 x 6 = 36). Dice probability often involves determining the likelihood of achieving a specific sum. For example, if we want the total of two dice to be 7, the possible combinations include the pairs:

• (1,6)
• (2,5)
• (3,4)
• (4,3)
• (5,2)
• (6,1)

This gives us 6 possible combinations. When analyzing dice probabilities, we look at favorable outcomes over possible outcomes. This basic principle helps us determine probabilities for other sums or specific conditions, such as one die showing a certain number given a sum.

Conditional Probability Formula

Conditional probability measures the likelihood of an event occurring given that another event has already occurred. This is what’s being evaluated when calculating probabilities like those in dice problems.
The conditional probability formula is given by: \[P(A|B) = \frac{P(A \cap B)}{P(B)}\]Where:

• P(A|B): The probability of event A occurring given that B is true.
• P(A \cap B): The probability of both A and B occurring.
• P(B): The probability of the event B.

In the exercise provided, the conditional probability of the first die showing 4, given that the sum is 7, was calculated as follows:- Identify the total occurrences of the event 'sum is 7' and find these combinations, which were 6.- The favorable condition (first die being 4) happens only once among these combinations.- Using the formula, the conditional probability was computed as \(\frac{1}{6}\). Understanding conditional probability is key for problems where one event influences another. It highlights dependencies between events, which is useful in games, predictive modeling, and risk assessment.