Sure! Let's break down the question and find step-by-step solutions for everything.
Given:
- [tex]\( P(T) = 0.4 \)[/tex]
- [tex]\( P(\operatorname{S \cap T}) = 0.5 \)[/tex]
- [tex]\( P(S \cup T) \)[/tex]
We need to find:
(a) [tex]\( P(S) \)[/tex]
(b) [tex]\( P\left(S' \cap T\right) \)[/tex]
### (a) Finding [tex]\( P(S) \)[/tex]
First, let's find [tex]\( P(S) \)[/tex].
We know from the given information that [tex]\( P(S \cap T) = 0.5 \)[/tex]. The probability [tex]\( P(S) \)[/tex] can be calculated via the relationship with the intersection probability. According to the problem statement, [tex]\( P(S \cap T) = 0.5 \)[/tex], and we also have:
[tex]\[ P(S \cap T) = P(S | T) \cdot P(T) \][/tex]
Here, [tex]\( P(S | T) \)[/tex] is the conditional probability of [tex]\( S \)[/tex] given [tex]\( T \)[/tex].
Given [tex]\( P(T) = 0.4 \)[/tex], we can rearrange the formula:
[tex]\[ P(S) = \frac{P(S \cap T)}{P(T)} \][/tex]
[tex]\[ P(S) = \frac{0.5}{0.4} \][/tex]
[tex]\[ P(S) = 1.25 \][/tex]
So,
[tex]\[ P(S) = 1.25 \][/tex]
### (b) Finding [tex]\( P\left(S' \cap T\right) \)[/tex]
Next, we need to find [tex]\( P\left(S' \cap T\right) \)[/tex].
Recall that [tex]\( S' \)[/tex] represents the complement of event [tex]\( S \)[/tex]. The probability of [tex]\( S' \cap T \)[/tex] can be calculated using the following relationship:
[tex]\[ P(T) = P(S \cap T) + P(S' \cap T) \][/tex]
From here, we can rearrange to find [tex]\( P(S' \cap T) \)[/tex]:
[tex]\[ P(S' \cap T) = P(T) - P(S \cap T) \][/tex]
Substituting the given values:
[tex]\[ P(S' \cap T) = 0.4 - 0.15 \][/tex]
[tex]\[ P(S' \cap T) = 0.25 \][/tex]
So,
[tex]\[ P\left(S' \cap T\right) = 0.25 \][/tex]
### Summary
(a) The probability of [tex]\( S \)[/tex], [tex]\( P(S) \)[/tex], is [tex]\( 1.25 \)[/tex].
(b) The probability of [tex]\( S' \cap T \)[/tex], [tex]\( P\left(S' \cap T\right) \)[/tex], is [tex]\( 0.25 \)[/tex].
I hope this solution helps clarify how to find the required probabilities step-by-step! If you have any further questions, feel free to ask.
