(ii) If the prohability of occurance oif at
least one of is 0.82 and
, find .

Ask by Boyd Barber. in Egypt

Jan 18,2025

Question

(ii) If the prohability of occurance oif at least one of $$ A, B $$ is 0.82 and $$ P(B-A)=0.26 $$, find $$ P(A) $$.

UpStudy AI · Accepted Answer

To determine $$ P(A) $$, let's analyze the given information step by step.

### Given:
1. **Probability of at least one of $$ A $$ or $$ B $$ occurring:**
   $$
   P(A \cup B) = 0.82
   $$
2. **Probability of $$ B $$ occurring without $$ A $$:**
   $$
   P(B - A) = P(B \cap A') = 0.26
   $$

### Objective:
Find $$ P(A) $$.

### Solution:

1. **Express $$ P(A \cup B) $$ using the principle of inclusion-exclusion:**
   $$
   P(A \cup B) = P(A) + P(B) - P(A \cap B)
   $$
   
2. **Express $$ P(B - A) $$ in terms of $$ P(B) $$ and $$ P(A \cap B) $$:**
   $$
   P(B - A) = P(B) - P(A \cap B) = 0.26
   $$
   $$
   \Rightarrow P(A \cap B) = P(B) - 0.26
   $$
   
3. **Substitute $$ P(A \cap B) $$ back into the inclusion-exclusion formula:**
   $$
   0.82 = P(A) + P(B) - (P(B) - 0.26)
   $$
   $$
   0.82 = P(A) + 0.26
   $$
   
4. **Solve for $$ P(A) $$:**
   $$
   P(A) = 0.82 - 0.26 = 0.56
   $$

### Conclusion:
The probability of event $$ A $$ occurring is **0.56**.

**Answer:**  
The probability of event A is 0.56.
$$ P(A) = 0.56 $$

(ii) If the prohability of occurance oif at
least one of is 0.82 and
, find .

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(ii) If the prohability of occurance oif at least one of is 0.82 and , find .