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.

  1. P(ANB) =
  2. P(AUB) =
  3. P(A |B) =
  4. 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:

  • P(A)=0.34P(A) = 0.34
  • P(B)=0.57P(B) = 0.57
  • Events AA and BB are independent.

For independent events, we know the following rules:

  • P(AB)=P(A)×P(B)P(A \cap B) = P(A) \times P(B) for independent events.
  • P(AB)=P(A)+P(B)P(AB)P(A \cup B) = P(A) + P(B) - P(A \cap B).
  • P(AB)=P(B)P(AB)P(A' \cap B) = P(B) - P(A \cap B).
  • P(AB)=P(AB)P(B)P(A | B) = \frac{P(A \cap B)}{P(B)}.

Now let's solve the problems one by one.


1. P(AB)P(A \cap B) (the intersection of A and B):

Since AA and BB are independent events, we can compute the intersection as: P(AB)=P(A)×P(B)=0.34×0.57=0.1938P(A \cap B) = P(A) \times P(B) = 0.34 \times 0.57 = 0.1938 So, P(AB)0.19P(A \cap B) \approx 0.19


2. P(AB)P(A \cup B) (the union of A and B):

The formula for the union of two events is: P(AB)=P(A)+P(B)P(AB)P(A \cup B) = P(A) + P(B) - P(A \cap B) Substitute the known values: P(AB)=0.34+0.570.19=0.910.19=0.72P(A \cup B) = 0.34 + 0.57 - 0.19 = 0.91 - 0.19 = 0.72 So, P(AB)=0.72P(A \cup B) = 0.72


3. P(AB)P(A | B) (the conditional probability of A given B):

The conditional probability formula is: P(AB)=P(AB)P(B)P(A | B) = \frac{P(A \cap B)}{P(B)} Substitute the known values: P(AB)=0.190.570.3333P(A | B) = \frac{0.19}{0.57} \approx 0.3333 So, P(AB)0.33P(A | B) \approx 0.33


4. P(AB)P(A' \cap B) (the intersection of the complement of A and B):

The complement of AA, denoted as AA', means the probability of not AA. The formula for this is: P(AB)=P(B)P(AB)P(A' \cap B) = P(B) - P(A \cap B) Substitute the known values: P(AB)=0.570.19=0.38P(A' \cap B) = 0.57 - 0.19 = 0.38 So, P(AB)=0.38P(A' \cap B) = 0.38


Summary of answers:

  1. P(AB)=0.19P(A \cap B) = 0.19
  2. P(AB)=0.72P(A \cup B) = 0.72
  3. P(AB)=0.33P(A | B) = 0.33
  4. P(AB)=0.38P(A' \cap B) = 0.38

Would you like more details on any of the steps? Here are five related questions for further exploration:

  1. How would the results change if AA and BB were not independent?
  2. Can you compute P(BA)P(B' \cap A)?
  3. What is the probability of ABA' \cup B'?
  4. How would you find the probability of P(AAB)P(A | A \cup B)?
  5. What would P(AB)P(A' | B) 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