Question
(ii) If the prohability of occurance oif at least one of \( A, B \) is 0.82 and \( P(B-A)=0.26 \), find \( P(A) \).
Ask by Boyd Barber. in Egypt
Jan 18,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( P(A) = 0.56 \)
Solution
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.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To find \( P(A) \), we can use the formula for the probability of the union of two events. We know that: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] From the problem, we know: \[ P(A \cup B) = 0.82 \] \[ P(B - A) = P(B) - P(A \cap B) = 0.26 \] We can denote \( P(A) \) as \( p \) and \( P(B) \) as \( q \). Therefore, we have: \[ q - P(A \cap B) = 0.26 \quad \text{(1)} \] Substituting \( P(A \cap B) \) with \( P(B) - 0.26 \) in the union formula, we obtain: \[ 0.82 = p + q - (q - 0.26) \] \[ 0.82 = p + 0.26 \] Rearranging gives us: \[ p = 0.82 - 0.26 = 0.56 \] Thus, the probability \( P(A) \) is: \[ \boxed{0.56} \]
