If A and B are two events such that P(A ∩ B)=0.1, and P(A | B) and P(B | A) are the roots of the equation 12 x²-7 x+1=0, then the value of P(Ā ∪ B̄)/P(Ā ∩ B̄) is :

Solution

Maths sir · 4 steps

Each conditional probability is 0.1 divided by one event's probability, so the roots give P(A) and P(B).

1. Solve the quadratic. 12x² - 7x + 1 = (4x - 1)(3x - 1) = 0, so the roots are 1/4 and 1/3.
2. Find P(A) and P(B). P(A | B) = 0.1/P(B) and P(B | A) = 0.1/P(A). Setting these equal to 1/4 and 1/3 makes P(A) and P(B) equal to 0.4 and 0.3, in some order. The order does not matter, because only their sum is used next.
3. Use De Morgan. P(A ∪ B) = 0.4 + 0.3 - 0.1 = 0.6. By De Morgan, Ā ∪ B̄ is the complement of A ∩ B, and Ā ∩ B̄ is the complement of A ∪ B.

   P(Ā ∪ B̄)/P(Ā ∩ B̄) = (1 - P(A ∩ B))/(1 - P(A ∪ B)) = (1 - 0.1)/(1 - 0.6) = 0.9/0.4

4. So. The ratio is 9/4, which is option (c).

The idea to keep. When conditional probabilities share the same intersection, divide that intersection by each one to get the single-event probabilities. Then De Morgan turns the complements into 1 - P(A ∩ B) and 1 - P(A ∪ B).

Answer from the NTA final answer key. Solution by PrepWiser.