document.write( "Question 1177191: If P(B) = p,P(A^c|B) = q, andP(A^c ∩ B^c) = r, find :
\n" ); document.write( "(a)P(A ∩ B^c),
\n" ); document.write( "(b)P(A),
\n" ); document.write( "(c)P(B|A).\r
\n" ); document.write( "\n" ); document.write( "thank you :)
\n" ); document.write( "

Algebra.Com's Answer #850504 by CPhill(1987)\"\" \"About 
You can put this solution on YOUR website!
Let's break down this problem step-by-step.\r
\n" ); document.write( "\n" ); document.write( "**Given:**\r
\n" ); document.write( "\n" ); document.write( "* P(B) = p
\n" ); document.write( "* P(Aᶜ|B) = q
\n" ); document.write( "* P(Aᶜ ∩ Bᶜ) = r\r
\n" ); document.write( "\n" ); document.write( "**Understanding the Notation:**\r
\n" ); document.write( "\n" ); document.write( "* Aᶜ: Complement of A (not A)
\n" ); document.write( "* Bᶜ: Complement of B (not B)
\n" ); document.write( "* A ∩ B: Intersection of A and B (A and B)
\n" ); document.write( "* A ∪ B: Union of A and B (A or B or both)
\n" ); document.write( "* P(A|B): Conditional probability of A given B\r
\n" ); document.write( "\n" ); document.write( "**Solving the Problem:**\r
\n" ); document.write( "\n" ); document.write( "**(a) P(A ∩ Bᶜ)**\r
\n" ); document.write( "\n" ); document.write( "1. **Use the Complement Rule:**
\n" ); document.write( " * P(Bᶜ) = 1 - P(B) = 1 - p\r
\n" ); document.write( "\n" ); document.write( "2. **Use De Morgan's Law:**
\n" ); document.write( " * (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ
\n" ); document.write( " * P((A ∪ B)ᶜ) = P(Aᶜ ∩ Bᶜ) = r\r
\n" ); document.write( "\n" ); document.write( "3. **Relate P(A ∪ B) to P(A ∩ Bᶜ):**
\n" ); document.write( " * P(A ∪ B) = 1 - P((A ∪ B)ᶜ) = 1 - r\r
\n" ); document.write( "\n" ); document.write( "4. **Use the Inclusion-Exclusion Principle:**
\n" ); document.write( " * P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
\n" ); document.write( " * 1 - r = P(A) + p - P(A ∩ B)\r
\n" ); document.write( "\n" ); document.write( "5. **Use the Conditional Probability P(Aᶜ|B) = q:**
\n" ); document.write( " * P(Aᶜ|B) = P(Aᶜ ∩ B) / P(B) = q
\n" ); document.write( " * P(Aᶜ ∩ B) = qP(B) = qp\r
\n" ); document.write( "\n" ); document.write( "6. **Use the Complement Rule for P(B):**
\n" ); document.write( " * P(B) = P(A ∩ B) + P(Aᶜ ∩ B)
\n" ); document.write( " * p = P(A ∩ B) + qp
\n" ); document.write( " * P(A ∩ B) = p - qp = p(1 - q)\r
\n" ); document.write( "\n" ); document.write( "7. **Find P(A ∩ Bᶜ):**
\n" ); document.write( " * P(A) = P(A ∩ B) + P(A ∩ Bᶜ)
\n" ); document.write( " * From step 4, P(A) = 1-r - p + P(A ∩ B)
\n" ); document.write( " * P(A) = 1-r-p + p(1-q)
\n" ); document.write( " * P(A) = 1-r-pq
\n" ); document.write( " * P(A ∩ B^c) = P(A) - P(A ∩ B)
\n" ); document.write( " * P(A ∩ B^c) = 1-r-pq - p(1-q)
\n" ); document.write( " * P(A ∩ B^c) = 1-r-pq - p + pq
\n" ); document.write( " * P(A ∩ B^c) = 1-r-p\r
\n" ); document.write( "\n" ); document.write( "**(b) P(A)**\r
\n" ); document.write( "\n" ); document.write( "* From step 7 in part (a):
\n" ); document.write( " * P(A) = 1 - r - pq\r
\n" ); document.write( "\n" ); document.write( "**(c) P(B|A)**\r
\n" ); document.write( "\n" ); document.write( "1. **Use the Conditional Probability Formula:**
\n" ); document.write( " * P(B|A) = P(A ∩ B) / P(A)\r
\n" ); document.write( "\n" ); document.write( "2. **Substitute the values:**
\n" ); document.write( " * P(B|A) = p(1 - q) / (1 - r - pq)\r
\n" ); document.write( "\n" ); document.write( "**Final Answers:**\r
\n" ); document.write( "\n" ); document.write( "* **(a) P(A ∩ Bᶜ) = 1 - r - p**
\n" ); document.write( "* **(b) P(A) = 1 - r - pq**
\n" ); document.write( "* **(c) P(B|A) = p(1 - q) / (1 - r - pq)**
\n" ); document.write( "
\n" ); document.write( "
\n" );