VIPSolutionsAct like a helpful tutor and exlain me :Q3. Probability with Independence Events A and B are independent such that P(A) = 0.4 and P(A UB) = 0.7. Find P(B).
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Act like a helpful tutor and exlain me :
Q3. Probability with Independence
Events A and B are independent such that
P(A) = 0.4 and P(A UB) = 0.7.
Find P(B).
This is AI generated Response.
Asked by: Alok Kumar
Created at: 2026-02-03 04:13:27
Asked by: Alok Kumar
Created at: 2026-02-03 04:13:27
Final Answer:
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## Step-by-Step Solution
### **Given:**
- \( P(A) = .4 \)
- \( P(A \cup B) = .7 \)
- **A and B are independent.**
### **Goal:**
Find \( P(B) \).
---
### **Step 1: Recall the Union Formula**
For any two events:
\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]
---
### **Step 2: Independence Definition**
For independent events:
\[
P(A \cap B) = P(A) \times P(B)
\]
---
### **Step 3: Substitute Values and Equations**
Plug values into the union formula:
\[
.7 = .4 + P(B) - [.4 \times P(B)]
\]
---
### **Step 4: Simplify Equation**
Let \( x = P(B) \):
\[
.7 = .4 + x - .4x
\]
\[
.7 = .4 + x(1 - .4)
\]
\[
.7 = .4 + .6x
\]
---
### **Step 5: Solve for \( x \) (i.e., \( P(B) \))**
\[
.7 - .4 = .6x
\]
\[
.3 = .6x
\]
\[
x = \frac{.3}{.6} = .5
\]
---
### **Final Answer**
\[
\boxed{P(B) = .5}
\]
---
**Summary:**
- Used the union formula and independence.
- Set up an equation with the given values.
- Solved for \( P(B) \).
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