Sure, let's solve the problem step-by-step given the conditions:
1. We are given the following probabilities:
- [tex]\( P(S \mid T) = 0.4 \)[/tex] (the probability of [tex]\( S \)[/tex] given [tex]\( T \)[/tex])
- [tex]\( P(S \cap T) = 0.15 \)[/tex] (the probability of the intersection of [tex]\( S \)[/tex] and [tex]\( T \)[/tex])
- [tex]\( P(S \mid \neg T) = 0.5 \)[/tex] (the probability of [tex]\( S \)[/tex] given the complement of [tex]\( T \)[/tex])
We need to determine:
1. [tex]\( P(S \cap T) \)[/tex]
2. [tex]\( P(S \cup T) \)[/tex]
3. [tex]\( P(S \mid T) \)[/tex]
Let's address each part individually:
1. Probability of the Intersection [tex]\( P(S \cap T) \)[/tex]:
This is directly provided as [tex]\( P(S \cap T) = 0.15 \)[/tex].
2. Probability of the Union [tex]\( P(S \cup T) \)[/tex]:
To find [tex]\( P(S \cup T) \)[/tex], we typically use the formula:
[tex]\[ P(S \cup T) = P(S) + P(T) - P(S \cap T) \][/tex]
However, in this case, we do not have the individual probabilities [tex]\( P(S) \)[/tex] and [tex]\( P(T) \)[/tex], so we cannot directly calculate [tex]\( P(S \cup T) \)[/tex].
Therefore, [tex]\( P(S \cup T) \)[/tex] cannot be determined with the given information.
3. Conditional Probability [tex]\( P(S \mid T) \)[/tex]:
This is directly provided as [tex]\( P(S \mid T) = 0.4 \)[/tex].
Thus, the solution to each part is:
1. [tex]\( P(S \cap T) = 0.15 \)[/tex]
2. [tex]\( P(S \cup T) = \text{Cannot be determined with given information} \)[/tex]
3. [tex]\( P(S \mid T) = 0.4 \)[/tex]
Hence, the answers can be summarized as:
[tex]\[ (0.15, \text{None}, 0.4) \][/tex]
