Westonci.ca is the trusted Q&A platform where you can get reliable answers from a community of knowledgeable contributors. Experience the convenience of getting accurate answers to your questions from a dedicated community of professionals. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
Alright, let's dive into the details of the problem to understand what is being asked and how to interpret the given probabilities.
### Understanding the Given Probabilities
We are given the following:
1. [tex]\( P(B_1) = 0.519 \)[/tex] - The probability of event [tex]\( B_1 \)[/tex] occurring.
2. [tex]\( P(B_1 \mid A_2) = 0.5820 \)[/tex] - The probability of event [tex]\( B_1 \)[/tex] occurring given that event [tex]\( A_2 \)[/tex] has occurred.
3. [tex]\( P(B_1 \mid A_1) > P(B_1) \)[/tex] - The probability of [tex]\( B_1 \)[/tex] occurring given [tex]\( A_1 \)[/tex] is greater than the probability of [tex]\( B_1 \)[/tex] alone.
4. [tex]\( P(B_1) = P(B_1 \mid A_2) \)[/tex] - The probability of [tex]\( B_1 \)[/tex] occurring is the same whether [tex]\( A_2 \)[/tex] has occurred or not.
### Concept of Independent Events
An event [tex]\( B_1 \)[/tex] and [tex]\( A_2 \)[/tex] are said to be independent if the occurrence of [tex]\( A_2 \)[/tex] does not affect the likelihood of [tex]\( B_1 \)[/tex] occurring. Mathematically, this can be represented as:
[tex]\[ P(B_1 \mid A_2) = P(B_1) \][/tex]
### Analysis of the Given Information
1. Comparing [tex]\( P(B_1) \)[/tex] and [tex]\( P(B_1 \mid A_2) \)[/tex]:
- We see that [tex]\( P(B_1 \mid A_2) = 0.5820 \)[/tex] and [tex]\( P(B_1) = 0.519 \)[/tex]. According to the given relationship [tex]\( P(B_1) = P(B_1 \mid A_2) \)[/tex], it suggests that the occurrence of [tex]\( A_2 \)[/tex] does not change the probability of [tex]\( B_1 \)[/tex]. This indicates that events [tex]\( A_2 \)[/tex] and [tex]\( B_1 \)[/tex] are independent.
2. Comparing [tex]\( P(B_1) \)[/tex] and [tex]\( P(B_1 \mid A_1) \)[/tex]:
- We are given that [tex]\( P(B_1 \mid A_1) > P(B_1) \)[/tex]. This means that knowing [tex]\( A_1 \)[/tex] has occurred makes [tex]\( B_1 \)[/tex] more likely to occur than knowing nothing about [tex]\( A_1 \)[/tex]. This indicates a dependence between [tex]\( A_1 \)[/tex] and [tex]\( B_1 \)[/tex].
### Conclusion
- Independence: The fact that [tex]\( P(B_1) = P(B_1 \mid A_2) \)[/tex] shows that [tex]\( B_1 \)[/tex] and [tex]\( A_2 \)[/tex] are independent events. There is no effect of [tex]\( A_2 \)[/tex] on the likelihood of [tex]\( B_1 \)[/tex] occurring.
- Dependence: The inequality [tex]\( P(B_1 \mid A_1) > P(B_1) \)[/tex] shows that [tex]\( B_1 \)[/tex] is more likely to occur when [tex]\( A_1 \)[/tex] has occurred. Consequently, [tex]\( A_1 \)[/tex] and [tex]\( B_1 \)[/tex] are not independent; the occurrence of [tex]\( A_1 \)[/tex] affects the probability of [tex]\( B_1 \)[/tex].
To summarize, the given probabilities demonstrate that [tex]\( B_1 \)[/tex] is independent of [tex]\( A_2 \)[/tex] but dependent on [tex]\( A_1 \)[/tex]. This highlights the importance of understanding the relationships between events in probability theory to make accurate assessments and predictions.
### Understanding the Given Probabilities
We are given the following:
1. [tex]\( P(B_1) = 0.519 \)[/tex] - The probability of event [tex]\( B_1 \)[/tex] occurring.
2. [tex]\( P(B_1 \mid A_2) = 0.5820 \)[/tex] - The probability of event [tex]\( B_1 \)[/tex] occurring given that event [tex]\( A_2 \)[/tex] has occurred.
3. [tex]\( P(B_1 \mid A_1) > P(B_1) \)[/tex] - The probability of [tex]\( B_1 \)[/tex] occurring given [tex]\( A_1 \)[/tex] is greater than the probability of [tex]\( B_1 \)[/tex] alone.
4. [tex]\( P(B_1) = P(B_1 \mid A_2) \)[/tex] - The probability of [tex]\( B_1 \)[/tex] occurring is the same whether [tex]\( A_2 \)[/tex] has occurred or not.
### Concept of Independent Events
An event [tex]\( B_1 \)[/tex] and [tex]\( A_2 \)[/tex] are said to be independent if the occurrence of [tex]\( A_2 \)[/tex] does not affect the likelihood of [tex]\( B_1 \)[/tex] occurring. Mathematically, this can be represented as:
[tex]\[ P(B_1 \mid A_2) = P(B_1) \][/tex]
### Analysis of the Given Information
1. Comparing [tex]\( P(B_1) \)[/tex] and [tex]\( P(B_1 \mid A_2) \)[/tex]:
- We see that [tex]\( P(B_1 \mid A_2) = 0.5820 \)[/tex] and [tex]\( P(B_1) = 0.519 \)[/tex]. According to the given relationship [tex]\( P(B_1) = P(B_1 \mid A_2) \)[/tex], it suggests that the occurrence of [tex]\( A_2 \)[/tex] does not change the probability of [tex]\( B_1 \)[/tex]. This indicates that events [tex]\( A_2 \)[/tex] and [tex]\( B_1 \)[/tex] are independent.
2. Comparing [tex]\( P(B_1) \)[/tex] and [tex]\( P(B_1 \mid A_1) \)[/tex]:
- We are given that [tex]\( P(B_1 \mid A_1) > P(B_1) \)[/tex]. This means that knowing [tex]\( A_1 \)[/tex] has occurred makes [tex]\( B_1 \)[/tex] more likely to occur than knowing nothing about [tex]\( A_1 \)[/tex]. This indicates a dependence between [tex]\( A_1 \)[/tex] and [tex]\( B_1 \)[/tex].
### Conclusion
- Independence: The fact that [tex]\( P(B_1) = P(B_1 \mid A_2) \)[/tex] shows that [tex]\( B_1 \)[/tex] and [tex]\( A_2 \)[/tex] are independent events. There is no effect of [tex]\( A_2 \)[/tex] on the likelihood of [tex]\( B_1 \)[/tex] occurring.
- Dependence: The inequality [tex]\( P(B_1 \mid A_1) > P(B_1) \)[/tex] shows that [tex]\( B_1 \)[/tex] is more likely to occur when [tex]\( A_1 \)[/tex] has occurred. Consequently, [tex]\( A_1 \)[/tex] and [tex]\( B_1 \)[/tex] are not independent; the occurrence of [tex]\( A_1 \)[/tex] affects the probability of [tex]\( B_1 \)[/tex].
To summarize, the given probabilities demonstrate that [tex]\( B_1 \)[/tex] is independent of [tex]\( A_2 \)[/tex] but dependent on [tex]\( A_1 \)[/tex]. This highlights the importance of understanding the relationships between events in probability theory to make accurate assessments and predictions.
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.