document.write( "Question 1177191: If P(B) = p,P(A^c|B) = q, andP(A^c ∩ B^c) = r, find :
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document.write( "(a)P(A ∩ B^c),
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document.write( "(b)P(A),
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document.write( "(c)P(B|A).\r
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document.write( "thank you :) \n" );
document.write( "
Algebra.Com's Answer #850504 by CPhill(1987)![]() ![]() 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( " |