Let `E_(1)` and `E_(2)` are the two independent events such that `P(E_(1))=0.35` and `P(E_(1) uu E_(2))=0.60`, find `P(E_(2))`.
A
`4/13`
B
`5/13`
C
`6/13`
D
`7/13`
Text Solution
AI Generated Solution
The correct Answer is:
To find \( P(E_2) \) given that \( P(E_1) = 0.35 \) and \( P(E_1 \cup E_2) = 0.60 \), we can use the formula for the probability of the union of two independent events.
### Step-by-Step Solution:
1. **Understand the Given Information**:
- We know that \( P(E_1) = 0.35 \).
- We also know that \( P(E_1 \cup E_2) = 0.60 \).
- The events \( E_1 \) and \( E_2 \) are independent.
2. **Use the Formula for the Union of Two Events**:
The formula for the probability of the union of two events is:
\[
P(E_1 \cup E_2) = P(E_1) + P(E_2) - P(E_1 \cap E_2)
\]
3. **Calculate \( P(E_1 \cap E_2) \)**:
Since \( E_1 \) and \( E_2 \) are independent, we have:
\[
P(E_1 \cap E_2) = P(E_1) \cdot P(E_2)
\]
Let \( P(E_2) = x \). Then:
\[
P(E_1 \cap E_2) = 0.35 \cdot x
\]
4. **Substitute into the Union Formula**:
Substitute \( P(E_1) \), \( P(E_1 \cup E_2) \), and \( P(E_1 \cap E_2) \) into the union formula:
\[
0.60 = 0.35 + x - (0.35 \cdot x)
\]
5. **Rearrange the Equation**:
Rearranging gives:
\[
0.60 = 0.35 + x - 0.35x
\]
Combine like terms:
\[
0.60 = 0.35 + (1 - 0.35)x
\]
\[
0.60 = 0.35 + 0.65x
\]
6. **Isolate \( x \)**:
Subtract \( 0.35 \) from both sides:
\[
0.60 - 0.35 = 0.65x
\]
\[
0.25 = 0.65x
\]
7. **Solve for \( x \)**:
Divide both sides by \( 0.65 \):
\[
x = \frac{0.25}{0.65} = \frac{25}{65} = \frac{5}{13}
\]
8. **Conclusion**:
Therefore, the probability of event \( E_2 \) is:
\[
P(E_2) = \frac{5}{13}
\]
To find \( P(E_2) \) given that \( P(E_1) = 0.35 \) and \( P(E_1 \cup E_2) = 0.60 \), we can use the formula for the probability of the union of two independent events.
### Step-by-Step Solution:
1. **Understand the Given Information**:
- We know that \( P(E_1) = 0.35 \).
- We also know that \( P(E_1 \cup E_2) = 0.60 \).
- The events \( E_1 \) and \( E_2 \) are independent.
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