Answer :
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.
