Math Problem Statement
For two events AA and BB, P(A)=0.3P(A)=0.3 and P(B)=0.1P(B)=0.1.
(a) If AA and BB are independent, then
P(A|B)P(A|B) ==
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P(A∪B)P(A∪B) ==
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P(A∩B)P(A∩B) ==
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(b) If AA and BB are dependent and P(A|B)=0.4P(A|B)=0.4, then
P(B|A)P(B|A) ==
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P(A∩B)P(A∩B) ==
Enter your answers exactly. Do not round.
Solution
Let's tackle each part of the problem step by step.
Part (a): Events and are independent
(i) :
For independent events and , the conditional probability is given by:
Since , we have:
(ii) :
For independent events, the probability of the union is given by:
Since , we have:
(iii) :
The probability of the intersection for independent events is:
Part (b): Events and are dependent with
(i) :
Using the conditional probability formula:
Given and :
So,
Now, use the conditional probability formula to find :
(ii) :
We already calculated this in the previous step:
Summary of Answers
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Part (a):
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Part (b):
Would you like any further details or have any questions? Here are some related questions you might find interesting:
- How do you calculate conditional probability?
- What is the significance of the independence of events in probability?
- How does dependence between events affect their probabilities?
- What are the key differences between union and intersection in probability?
- How does Bayes' theorem relate to conditional probabilities?
Tip: When working with conditional probabilities, ensure you clearly understand whether events are independent or dependent, as it significantly affects the calculations.
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Math Problem Analysis
Mathematical Concepts
Probability
Conditional Probability
Independence and Dependence of Events
Formulas
Conditional Probability formula
Probability of Union formula
Probability of Intersection formula
Theorems
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Suitable Grade Level
College Level
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Examples and Exercises on Probability: Independent and Dependent Events