What Is The Probability Of The Complement Of Rolling A Number Less Than 5 Using A Six-sided Die?A. { \frac{1}{6}$}$ B. { \frac{1}{3}$}$ C. { \frac{2}{5}$}$ D. { \frac{2}{3}$}$
Introduction
Probability is a fundamental concept in mathematics that deals with the study of chance events. It is a measure of the likelihood of an event occurring, and it is often expressed as a number between 0 and 1. In this article, we will explore the concept of probability and its application to a specific problem: finding the probability of the complement of rolling a number less than 5 using a six-sided die.
What is Probability?
Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1 that represents the chance of an event happening. The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
The Basics of Probability Notation
Probability is often denoted by the letter P. The probability of an event A is written as P(A). The probability of the complement of an event A, denoted by A', is written as P(A').
The Problem: Rolling a Number Less Than 5
The problem we are trying to solve is finding the probability of the complement of rolling a number less than 5 using a six-sided die. To solve this problem, we need to first find the probability of rolling a number less than 5.
Step 1: Define the Sample Space
The sample space is the set of all possible outcomes of an experiment. In this case, the sample space consists of the numbers 1, 2, 3, 4, 5, and 6.
Step 2: Count the Number of Favorable Outcomes
The number of favorable outcomes is the number of outcomes that satisfy the condition of rolling a number less than 5. In this case, the favorable outcomes are the numbers 1, 2, 3, and 4.
Step 3: Count the Total Number of Possible Outcomes
The total number of possible outcomes is the total number of outcomes in the sample space. In this case, the total number of possible outcomes is 6.
Step 4: Calculate the Probability
The probability of rolling a number less than 5 is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
P(rolling a number less than 5) = Number of favorable outcomes / Total number of possible outcomes = 4 / 6 = 2/3
The Complement of Rolling a Number Less Than 5
The complement of rolling a number less than 5 is rolling a number greater than or equal to 5. This includes the outcomes 5 and 6.
Step 1: Count the Number of Favorable Outcomes
The number of favorable outcomes is the number of outcomes that satisfy the condition of rolling a number greater than or equal to 5. In this case, the favorable outcomes are the numbers 5 and 6.
Step 2: Count the Total Number of Possible Outcomes
The total number of possible outcomes is the total number of outcomes in the sample space. In this case, the total number of possible outcomes is 6.
Step 3: Calculate the Probability
The probability of rolling a number greater than or equal to 5 is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
P(rolling a number greater than or equal to 5) = Number of favorable outcomes / Total number of possible outcomes = 2 / 6 = 1/3
Conclusion
In conclusion, the probability of rolling a number less than 5 using a six-sided die is 2/3. The probability of the complement of rolling a number less than 5, which is rolling a number greater than or equal to 5, is 1/3.
Answer
The correct answer is D. .
Final Thoughts
Q: What is the probability of rolling a number greater than 4 using a six-sided die?
A: The probability of rolling a number greater than 4 using a six-sided die is 1/3. This is because there are 2 favorable outcomes (5 and 6) out of a total of 6 possible outcomes.
Q: What is the complement of rolling a number greater than 4?
A: The complement of rolling a number greater than 4 is rolling a number less than or equal to 4. This includes the outcomes 1, 2, 3, and 4.
Q: What is the probability of rolling a number less than or equal to 4 using a six-sided die?
A: The probability of rolling a number less than or equal to 4 using a six-sided die is 2/3. This is because there are 4 favorable outcomes (1, 2, 3, and 4) out of a total of 6 possible outcomes.
Q: What is the relationship between the probability of an event and its complement?
A: The probability of an event and its complement are complementary probabilities. This means that the sum of the probability of an event and the probability of its complement is equal to 1.
Q: How do you calculate the probability of the complement of an event?
A: To calculate the probability of the complement of an event, you need to first calculate the probability of the event itself. Then, you subtract the probability of the event from 1 to get the probability of the complement.
Q: What is the probability of rolling a number greater than 3 using a six-sided die?
A: The probability of rolling a number greater than 3 using a six-sided die is 2/3. This is because there are 3 favorable outcomes (4, 5, and 6) out of a total of 6 possible outcomes.
Q: What is the complement of rolling a number greater than 3?
A: The complement of rolling a number greater than 3 is rolling a number less than or equal to 3. This includes the outcomes 1, 2, and 3.
Q: What is the probability of rolling a number less than or equal to 3 using a six-sided die?
A: The probability of rolling a number less than or equal to 3 using a six-sided die is 1/2. This is because there are 3 favorable outcomes (1, 2, and 3) out of a total of 6 possible outcomes.
Q: Can you give an example of a situation where the probability of an event and its complement are equal?
A: Yes, one example is rolling a fair coin. The probability of getting heads is 1/2, and the probability of getting tails is also 1/2. In this case, the probability of the event (getting heads) and its complement (getting tails) are equal.
Q: What is the probability of rolling a number less than 2 using a six-sided die?
A: The probability of rolling a number less than 2 using a six-sided die is 1/3. This is because there is 1 favorable outcome (1) out of a total of 6 possible outcomes.
Q: What is the complement of rolling a number less than 2?
A: The complement of rolling a number less than 2 is rolling a number greater than or equal to 2. This includes the outcomes 2, 3, 4, 5, and 6.
Q: What is the probability of rolling a number greater than or equal to 2 using a six-sided die?
A: The probability of rolling a number greater than or equal to 2 using a six-sided die is 5/6. This is because there are 5 favorable outcomes (2, 3, 4, 5, and 6) out of a total of 6 possible outcomes.