Math Problem Statement
Consider the sample space Sequals={o 1 comma o 2 comma o 3 comma o 4o1, o2, o3, o4}. Suppose that Pr left parenthesis o 1 right parenthesis plus Pr left parenthesis o 3 right parenthesis equals Pr left parenthesis o 2 right parenthesis plus Pr left parenthesis o 4 right parenthesisPro1+Pro3=Pro2+Pro4 and that Pr left parenthesis o 1 right parenthesis equals 0.07Pro1=0.07. (a) Find the probability assignment for the probability space when o 2 and o 3o2 and o3 have the same probability. (b) Find the probability assignment for the probability space when Pr left parenthesis o 2 right parenthesis equals Pro2=0.17
Solution
Given the problem, we have the sample space and the following conditions:
We are asked to solve for the probability assignments under two scenarios.
Part (a): When
Let and .
Using the first condition:
Substituting the given values:
Simplifying this:
Thus, .
Now, the sum of all probabilities must equal 1:
Substituting the known values:
Therefore, the probability assignments are:
Part (b): When
Let and .
Using the first condition:
Substituting the given values:
Simplifying:
Now, the sum of all probabilities must equal 1:
Substituting the known values:
Now, solving the system of equations:
- From Equation 1:
- Substituting into Equation 2:
Thus, .
Therefore, the probability assignments are:
Would you like further details or have any questions?
Here are some related questions you might consider:
- How would the probability distribution change if was given instead of ?
- What are the possible values of if ?
- Can the conditions be satisfied if was greater than 0.5?
- What is the relationship between the probabilities if ?
- How would the probability assignment change if the sample space had 5 elements instead of 4?
Tip: When solving probability problems, always ensure the sum of all probabilities equals 1 to verify your calculations.
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Math Problem Analysis
Mathematical Concepts
Probability
Equations
Sample Space
Formulas
Pr(o1) + Pr(o3) = Pr(o2) + Pr(o4)
Pr(o1) + Pr(o2) + Pr(o3) + Pr(o4) = 1
Theorems
Law of Total Probability
Suitable Grade Level
Grades 10-12
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