Math Problem Statement
Suppose A and B are events such that P(A) = 0.34 and P(B) = 0.57 . Assuming that events A and B are independent, help Kori answer the following questions. You may round your answers to 2 decimal places, and use previous values in subsequent computations. Showing work is REQUIRED.
- P(ANB) =
- P(AUB) =
- P(A |B) =
- P(A' В) =
Solution
Let's go through these step by step, using the provided values and showing the necessary work for each part.
Given:
- Events and are independent.
For independent events, we know the following rules:
- for independent events.
- .
- .
- .
Now let's solve the problems one by one.
1. (the intersection of A and B):
Since and are independent events, we can compute the intersection as: So,
2. (the union of A and B):
The formula for the union of two events is: Substitute the known values: So,
3. (the conditional probability of A given B):
The conditional probability formula is: Substitute the known values: So,
4. (the intersection of the complement of A and B):
The complement of , denoted as , means the probability of not . The formula for this is: Substitute the known values: So,
Summary of answers:
Would you like more details on any of the steps? Here are five related questions for further exploration:
- How would the results change if and were not independent?
- Can you compute ?
- What is the probability of ?
- How would you find the probability of ?
- What would represent and how would you calculate it?
Tip: Remember that conditional probabilities can be interpreted as the probability of one event happening, given that another has occurred. It's a key concept in understanding dependent relationships between events!
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Math Problem Analysis
Mathematical Concepts
Probability
Independent Events
Conditional Probability
Formulas
P(A ∩ B) = P(A) × P(B) for independent events
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
P(A | B) = P(A ∩ B) / P(B)
P(A' ∩ B) = P(B) - P(A ∩ B)
Theorems
Multiplication Rule for Independent Events
Addition Rule for Probabilities
Conditional Probability Theorem
Suitable Grade Level
Grades 10-12
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