Find solutions to your questions with the help of IDNLearn.com's expert community. Discover reliable answers to your questions with our extensive database of expert knowledge.
Sagot :
To determine the correct condition that must be true when events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent, let’s start by revisiting the definition of independent events in probability.
### Definition of Independent Events
Two events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are said to be independent if the occurrence of one event does not affect the probability of the other event occurring. Mathematically, this is expressed as:
[tex]\[ P(A \cap B) = P(A) \cdot P(B) \][/tex]
Additionally, for independent events [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ P(A \mid B) = P(A) \][/tex]
[tex]\[ P(B \mid A) = P(B) \][/tex]
### Analyzing Each Option
Let’s analyze each option given in the question:
A. [tex]\( P(A \mid B) = x \)[/tex]
Since [tex]\( x \)[/tex] represents [tex]\( P(A) \)[/tex] and knowing that [tex]\( P(A \mid B) = P(A) \)[/tex], we can rewrite this as:
[tex]\[ P(A \mid B) = P(A) \][/tex]
Thus, this condition simplifies to:
[tex]\[ P(A \mid B) = x \][/tex]
This condition is true for independent events.
B. [tex]\( P(A \mid B) = y \)[/tex]
Here, [tex]\( y \)[/tex] represents [tex]\( P(B) \)[/tex]. For independent events, [tex]\( P(A \mid B) \)[/tex] should equal [tex]\( P(A) \)[/tex] and not [tex]\( P(B) \)[/tex]. Therefore:
[tex]\[ P(A \mid B) \neq P(B) \][/tex]
This condition is false for independent events.
C. [tex]\( P(B \mid A) = x \)[/tex]
In this case, [tex]\( x \)[/tex] still represents [tex]\( P(A) \)[/tex]. For independent events, [tex]\( P(B \mid A) \)[/tex] should equal [tex]\( P(B) \)[/tex] and not [tex]\( P(A) \)[/tex]. Thus:
[tex]\[ P(B \mid A) \neq P(A) \][/tex]
This condition is false for independent events.
D. [tex]\( P(B \mid A) = xy \)[/tex]
This implies that [tex]\( P(B \mid A) \)[/tex] is equal to the product of [tex]\( P(A) \)[/tex] and [tex]\( P(B) \)[/tex]. For independent events, we know:
[tex]\[ P(B \mid A) = P(B) \][/tex]
Therefore, equating it to [tex]\( xy \)[/tex] would not be correct:
[tex]\[ P(B \mid A) \neq P(A) \cdot P(B) \][/tex]
This condition is false for independent events.
### Conclusion
Based on the analysis above, the correct condition that must be true when events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent is:
A. [tex]\( P(A \mid B) = x \)[/tex]
### Definition of Independent Events
Two events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are said to be independent if the occurrence of one event does not affect the probability of the other event occurring. Mathematically, this is expressed as:
[tex]\[ P(A \cap B) = P(A) \cdot P(B) \][/tex]
Additionally, for independent events [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ P(A \mid B) = P(A) \][/tex]
[tex]\[ P(B \mid A) = P(B) \][/tex]
### Analyzing Each Option
Let’s analyze each option given in the question:
A. [tex]\( P(A \mid B) = x \)[/tex]
Since [tex]\( x \)[/tex] represents [tex]\( P(A) \)[/tex] and knowing that [tex]\( P(A \mid B) = P(A) \)[/tex], we can rewrite this as:
[tex]\[ P(A \mid B) = P(A) \][/tex]
Thus, this condition simplifies to:
[tex]\[ P(A \mid B) = x \][/tex]
This condition is true for independent events.
B. [tex]\( P(A \mid B) = y \)[/tex]
Here, [tex]\( y \)[/tex] represents [tex]\( P(B) \)[/tex]. For independent events, [tex]\( P(A \mid B) \)[/tex] should equal [tex]\( P(A) \)[/tex] and not [tex]\( P(B) \)[/tex]. Therefore:
[tex]\[ P(A \mid B) \neq P(B) \][/tex]
This condition is false for independent events.
C. [tex]\( P(B \mid A) = x \)[/tex]
In this case, [tex]\( x \)[/tex] still represents [tex]\( P(A) \)[/tex]. For independent events, [tex]\( P(B \mid A) \)[/tex] should equal [tex]\( P(B) \)[/tex] and not [tex]\( P(A) \)[/tex]. Thus:
[tex]\[ P(B \mid A) \neq P(A) \][/tex]
This condition is false for independent events.
D. [tex]\( P(B \mid A) = xy \)[/tex]
This implies that [tex]\( P(B \mid A) \)[/tex] is equal to the product of [tex]\( P(A) \)[/tex] and [tex]\( P(B) \)[/tex]. For independent events, we know:
[tex]\[ P(B \mid A) = P(B) \][/tex]
Therefore, equating it to [tex]\( xy \)[/tex] would not be correct:
[tex]\[ P(B \mid A) \neq P(A) \cdot P(B) \][/tex]
This condition is false for independent events.
### Conclusion
Based on the analysis above, the correct condition that must be true when events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent is:
A. [tex]\( P(A \mid B) = x \)[/tex]
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Find clear answers at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.
