When rolling a fair, six-sided number cube, what is the probability of rolling an even number or a number less than 3?
When rolling a fair, six-sided number cube, what is the probability of rolling an even number or a number less than 3?
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
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![### Probability Question on Rolling a Number Cube
**Question:**
When rolling a fair, six-sided number cube, what is the probability of rolling an even number or a number less than 3?
**Answer Choices:**
- a) \(\frac{5}{6}\)
- b) \(\frac{1}{2}\)
- c) \(\frac{1}{3}\)
- d) \(\frac{2}{3}\)
**Solution Explanation:**
To solve this probability question, consider the numbers on a six-sided number cube, which are {1, 2, 3, 4, 5, 6}.
1. **Even Numbers:**
The even numbers on a dice are 2, 4, and 6, so there are 3 even numbers.
2. **Numbers Less Than 3:**
The numbers on the dice that are less than 3 are 1 and 2, so there are 2 numbers less than 3.
However, there's an overlap in these two sets (number 2), and it shouldn't be counted twice.
3. **Total Outcomes:**
- Even numbers: 2, 4, 6 (3 outcomes)
- Less than 3: 1, 2 (2 outcomes)
- Overlap: 2 (1 outcome)
The probability can be calculated using the formula for the union of two sets:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B)\]
Where:
- \(P(A)\) is the probability of selecting an even number = \(\frac{3}{6} = \frac{1}{2}\)
- \(P(B)\) is the probability of selecting a number less than 3 = \(\frac{2}{6} = \frac{1}{3}\)
- \(P(A \cap B)\) is the probability of selecting a number that is both even and less than 3 = \(\frac{1}{6}\)
So, the probability is:
\[ P(A \cup B) = \frac{1}{2} + \frac{1}{3} - \frac{1}{6}\]
Finding a common denominator (6), we get:
\[ P(A \cup B) = \frac{3}{6} + \frac{2}{6} - \frac{1}{6](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F636011de-a5ad-4171-a296-99fd2e0537ff%2F6a25033b-11c2-423d-9dac-422746c34091%2Fmqrswse_processed.png&w=3840&q=75)
Transcribed Image Text:### Probability Question on Rolling a Number Cube
**Question:**
When rolling a fair, six-sided number cube, what is the probability of rolling an even number or a number less than 3?
**Answer Choices:**
- a) \(\frac{5}{6}\)
- b) \(\frac{1}{2}\)
- c) \(\frac{1}{3}\)
- d) \(\frac{2}{3}\)
**Solution Explanation:**
To solve this probability question, consider the numbers on a six-sided number cube, which are {1, 2, 3, 4, 5, 6}.
1. **Even Numbers:**
The even numbers on a dice are 2, 4, and 6, so there are 3 even numbers.
2. **Numbers Less Than 3:**
The numbers on the dice that are less than 3 are 1 and 2, so there are 2 numbers less than 3.
However, there's an overlap in these two sets (number 2), and it shouldn't be counted twice.
3. **Total Outcomes:**
- Even numbers: 2, 4, 6 (3 outcomes)
- Less than 3: 1, 2 (2 outcomes)
- Overlap: 2 (1 outcome)
The probability can be calculated using the formula for the union of two sets:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B)\]
Where:
- \(P(A)\) is the probability of selecting an even number = \(\frac{3}{6} = \frac{1}{2}\)
- \(P(B)\) is the probability of selecting a number less than 3 = \(\frac{2}{6} = \frac{1}{3}\)
- \(P(A \cap B)\) is the probability of selecting a number that is both even and less than 3 = \(\frac{1}{6}\)
So, the probability is:
\[ P(A \cup B) = \frac{1}{2} + \frac{1}{3} - \frac{1}{6}\]
Finding a common denominator (6), we get:
\[ P(A \cup B) = \frac{3}{6} + \frac{2}{6} - \frac{1}{6
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