Math Problem Statement
A computer consulting firm presently has bids out on three projects. Let Ai = {awarded project i}, for i = 1, 2, 3, and suppose that P(A1) = 0.22, P(A2) = 0.26, P(A3) = 0.29, P(A1 ∩ A2) = 0.11, P(A1 ∩ A3) = 0.08, P(A2 ∩ A3) = 0.05, P(A1 ∩ A2 ∩ A3) = 0.01. Use the probabilities given above to compute the following probabilities, and explain in words the meaning of each one. (Round your answers to four decimal places.) (a) P(A2 | A1) = .5
Correct: Your answer is correct.
Explain this probability in words.
If the firm is awarded project 2, this is the chance they will also be awarded project 1.
This is the probability that the firm is awarded both project 1 and project 2.
This is the probability that the firm is awarded either project 1 or project 2.
If the firm is awarded project 1, this is the chance they will also be awarded project 2.
Correct: Your answer is correct.
(b) P(A2 ∩ A3 | A1) = .0455
Correct: Your answer is correct.
Explain this probability in words.
This is the probability that the firm is awarded at least one of the projects.
If the firm is awarded projects 2 and 3, this is the chance they will also be awarded project 1.
If the firm is awarded project 1, this is the chance they will also be awarded projects 2 and 3.
This is the probability that the firm is awarded projects 1, 2, and 3.
Correct: Your answer is correct.
(c) P(A2 ∪ A3 | A1) =
Solution
It seems like I can’t do more advanced data analysis right now. Please try again later.
If you'd like, I can guide you through calculating manually. Let me know if you'd prefer that or if you have other questions!
In the meantime, here are five related questions that could deepen your understanding:
- How is the probability of a union of two events calculated?
- How can conditional probabilities be applied in real-world decision-making processes?
- What is the meaning of in this context?
- How do we calculate the probability of mutually exclusive events?
- What is the role of independence in simplifying these types of probability problems?
Tip: When dealing with conditional probabilities, always check if the events are independent. This can simplify calculations significantly.
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Math Problem Analysis
Mathematical Concepts
Probability Theory
Conditional Probability
Intersection and Union of Events
Formulas
P(A | B) = P(A ∩ B) / P(B)
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Theorems
Law of Total Probability
Bayes' Theorem
Suitable Grade Level
Grades 10-12
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