assume that all the odd numbers on a weighted die are equally likely and all the even numbers are equally likely, the odd numvers are k times as likely as the even numbers and Pr[2]=1/21 what is the value of k

Let's solve this step-by-step.

1. **Define the problem and probabilities:**

   - The die has six faces with numbers \( 1, 2, 3, 4, 5, \) and \( 6 \).
   - The odd numbers are \( 1, 3, 5 \).
   - The even numbers are \( 2, 4, 6 \).
   - The probability of all odd numbers is the same, and let it be denoted as \( p \).
   - The probability of all even numbers is the same, and let it be denoted as \( q \).
   - We know that \( p = k \times q \) because the probability of odd numbers is \( k \) times that of even numbers.
   - It is given that \( \Pr[2] = q = \frac{1}{21} \).

2. **Sum of probabilities:**

   Since the die is fair, the sum of all probabilities must be 1:

   \[
   3p + 3q = 1
   \]

   Substitute \( p = kq \) into the equation:

   \[
   3(kq) + 3q = 1
   \]

   Factor out \( q \):

   \[
   3q(k + 1) = 1
   \]

   Substitute \( q = \frac{1}{21} \):

   \[
   3 \times \frac{1}{21} \times (k + 1) = 1
   \]

3. **Solve for \( k \):**

   Simplify the equation:

   \[
   \frac{3(k + 1)}{21} = 1
   \]

   \[
   \frac{k + 1}{7} = 1
   \]

   \[
   k + 1 = 7
   \]

   \[
   k = 6
   \]

Therefore, the value of \( k \) is \( 6 \).

Do you want more details on any part of the solution? 

### Relative Questions
1. What is the probability of rolling an odd number on this die?
2. How does the value of \( k \) change if the probability of rolling a 2 is altered?
3. What is the probability of rolling a number greater than 3?
4. Can we apply the same method for a die with more than six sides?
5. How would the probability distribution look if all numbers were equally likely?

### Tip
When dealing with probability problems involving conditions, always start by defining variables for the unknown probabilities and sum them to 1.

Assume that all the odd numbers on a weighted die are equally likely and all the even numbers are equally likely. The odd numbers are k times as likely as the even numbers, and Pr[2] = 1/21. What is the value of k?

Learn how to calculate the value of k in a weighted die where odd numbers are k times more likely than even numbers, and the probability of rolling a 2 is 1/21. This solution involves key probability concepts and step-by-step algebraic manipulation. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation