MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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**Title: Calculating Conditional Probability with a Six-Sided Die** **Introduction:** In this lesson, we will explore how to calculate conditional probabilities using a fair six-sided die. We will define two specific events and determine the probability of one event given that another has occurred. **Problem Statement:** A fair six-sided die is rolled. Define the following events: - Event A: The outcome is less than 4. - Event B: The outcome is an odd number. **Objective:** Find the conditional probability P(A|B), the probability of event A occurring given that event B has occurred. **Analysis:** To solve this problem, we start by considering the possible outcomes: - The outcomes of rolling a six-sided die are 1, 2, 3, 4, 5, and 6. - Event A includes the outcomes: 1, 2, 3. - Event B includes the outcomes: 1, 3, 5. **Solution Steps:** 1. Identify the outcomes that satisfy both events A and B: these are the odd numbers less than 4. Thus, the outcomes are 1 and 3. 2. Calculate the size of the sample space for event B: three outcomes (1, 3, 5). 3. Calculate the number of favorable outcomes for both A and B: two outcomes (1, 3). 4. Use the conditional probability formula: \[ P(A|B) = \frac{\text{Number of outcomes in both A and B}}{\text{Number of outcomes in B}} \] 5. Thus, \( P(A|B) = \frac{2}{3} \). **Conclusion:** The probability of rolling a number less than 4 given that the number is odd is \(\frac{2}{3}\). **Note:** The handwritten list on the page notes the fractions representing equivalent probabilities, though not explicitly solved in steps, they provide a reference for calculating probabilities with fractions of the total possible outcomes (e.g., 1/6, 2/6, etc.).
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Transcribed Image Text:**Title: Calculating Conditional Probability with a Six-Sided Die** **Introduction:** In this lesson, we will explore how to calculate conditional probabilities using a fair six-sided die. We will define two specific events and determine the probability of one event given that another has occurred. **Problem Statement:** A fair six-sided die is rolled. Define the following events: - Event A: The outcome is less than 4. - Event B: The outcome is an odd number. **Objective:** Find the conditional probability P(A|B), the probability of event A occurring given that event B has occurred. **Analysis:** To solve this problem, we start by considering the possible outcomes: - The outcomes of rolling a six-sided die are 1, 2, 3, 4, 5, and 6. - Event A includes the outcomes: 1, 2, 3. - Event B includes the outcomes: 1, 3, 5. **Solution Steps:** 1. Identify the outcomes that satisfy both events A and B: these are the odd numbers less than 4. Thus, the outcomes are 1 and 3. 2. Calculate the size of the sample space for event B: three outcomes (1, 3, 5). 3. Calculate the number of favorable outcomes for both A and B: two outcomes (1, 3). 4. Use the conditional probability formula: \[ P(A|B) = \frac{\text{Number of outcomes in both A and B}}{\text{Number of outcomes in B}} \] 5. Thus, \( P(A|B) = \frac{2}{3} \). **Conclusion:** The probability of rolling a number less than 4 given that the number is odd is \(\frac{2}{3}\). **Note:** The handwritten list on the page notes the fractions representing equivalent probabilities, though not explicitly solved in steps, they provide a reference for calculating probabilities with fractions of the total possible outcomes (e.g., 1/6, 2/6, etc.).
Expert Solution
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Step 1: Determine the given data in the question

The experiment is of rolling a fair die.

A fair die has the following outcomes 1 comma 2 comma 3 comma 4 comma 5 comma 6.

The outcomes are equally likely.

So probability for each outcome is 1 over 6.

To find the probability for given events.


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