Experience the convenience of getting your questions answered at IDNLearn.com. Our experts are ready to provide prompt and detailed answers to any questions you may have.
Sagot :
To solve this problem, we need to understand the concept of independent events in probability theory.
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 occurrence of the other event. Mathematically, if [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent events, the following condition holds true:
[tex]\[ P(A \mid B) = P(A) \][/tex]
This means that the probability of event [tex]\( A \)[/tex] occurring given that event [tex]\( B \)[/tex] has occurred is simply the probability of event [tex]\( A \)[/tex] occurring, unaffected by [tex]\( B \)[/tex].
Given this understanding, let's analyze the options provided:
- Option A: [tex]\( P(A \mid B)=y \)[/tex]
- This option suggests that the probability of [tex]\( A \)[/tex] given [tex]\( B \)[/tex] equals the probability of [tex]\( B \)[/tex]. This does not align with the definition of independence, which relates [tex]\( P(A \mid B) \)[/tex] to [tex]\( P(A) \)[/tex], not [tex]\( P(B) \)[/tex].
- Option B: [tex]\( P(B \mid A)=x \)[/tex]
- This option suggests that the probability of [tex]\( B \)[/tex] given [tex]\( A \)[/tex] equals the probability of [tex]\( A \)[/tex]. This does not align with the definition of independence. The correct relation for independent events would involve comparing [tex]\( P(B \mid A) \)[/tex] to [tex]\( P(B) \)[/tex].
- Option C: [tex]\( P(A \mid B)=x \)[/tex]
- This option is in line with the definition of independence. It directly states that [tex]\( P(A \mid B) \)[/tex] equals [tex]\( x \)[/tex], which is the same as [tex]\( P(A) \)[/tex]. This matches the condition for independent events where [tex]\( P(A \mid B) = P(A) \)[/tex].
- Option D: [tex]\( P(B \mid A)=x y \)[/tex]
- This option suggests that [tex]\( P(B \mid A) \)[/tex] is the product of [tex]\( x \)[/tex] and [tex]\( y \)[/tex], which does not relate correctly to the definition of independence. The independence condition requires that [tex]\( P(A \mid B) = P(A) \)[/tex] and [tex]\( P(B \mid A) = P(B) \)[/tex], without introducing the product of [tex]\( P(A) \)[/tex] and [tex]\( P(B) \)[/tex].
Therefore, based on the condition for the independence of events, the correct answer is Option C: [tex]\( P(A \mid B)=x \)[/tex]. This satisfies the requirement that [tex]\( P(A \mid B) = P(A) \)[/tex] for 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 occurrence of the other event. Mathematically, if [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are independent events, the following condition holds true:
[tex]\[ P(A \mid B) = P(A) \][/tex]
This means that the probability of event [tex]\( A \)[/tex] occurring given that event [tex]\( B \)[/tex] has occurred is simply the probability of event [tex]\( A \)[/tex] occurring, unaffected by [tex]\( B \)[/tex].
Given this understanding, let's analyze the options provided:
- Option A: [tex]\( P(A \mid B)=y \)[/tex]
- This option suggests that the probability of [tex]\( A \)[/tex] given [tex]\( B \)[/tex] equals the probability of [tex]\( B \)[/tex]. This does not align with the definition of independence, which relates [tex]\( P(A \mid B) \)[/tex] to [tex]\( P(A) \)[/tex], not [tex]\( P(B) \)[/tex].
- Option B: [tex]\( P(B \mid A)=x \)[/tex]
- This option suggests that the probability of [tex]\( B \)[/tex] given [tex]\( A \)[/tex] equals the probability of [tex]\( A \)[/tex]. This does not align with the definition of independence. The correct relation for independent events would involve comparing [tex]\( P(B \mid A) \)[/tex] to [tex]\( P(B) \)[/tex].
- Option C: [tex]\( P(A \mid B)=x \)[/tex]
- This option is in line with the definition of independence. It directly states that [tex]\( P(A \mid B) \)[/tex] equals [tex]\( x \)[/tex], which is the same as [tex]\( P(A) \)[/tex]. This matches the condition for independent events where [tex]\( P(A \mid B) = P(A) \)[/tex].
- Option D: [tex]\( P(B \mid A)=x y \)[/tex]
- This option suggests that [tex]\( P(B \mid A) \)[/tex] is the product of [tex]\( x \)[/tex] and [tex]\( y \)[/tex], which does not relate correctly to the definition of independence. The independence condition requires that [tex]\( P(A \mid B) = P(A) \)[/tex] and [tex]\( P(B \mid A) = P(B) \)[/tex], without introducing the product of [tex]\( P(A) \)[/tex] and [tex]\( P(B) \)[/tex].
Therefore, based on the condition for the independence of events, the correct answer is Option C: [tex]\( P(A \mid B)=x \)[/tex]. This satisfies the requirement that [tex]\( P(A \mid B) = P(A) \)[/tex] for independent events.
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. IDNLearn.com is committed to providing accurate answers. Thanks for stopping by, and see you next time for more solutions.
