A and B are two events such that P(A ∩ B) = 0.1 P(A' ∩ B') = 0.2 P(B) = 2P(A) (a) Find P(A) (b) Find P(B | A) - AQA - A-Level Maths Mechanics - Question 14 - 2021 - Paper 3

To find P(A), we start by using the given probabilities:

1. We know that:

   • P(A ∩ B) = 0.1
   • P(A' ∩ B') = 0.2
   • P(B) = 2P(A)

2. Using the total probability rule, we have: $P(A) + P(B) + P(A' ∩ B') = 1$ where P(A) + P(B) + P(A' ∩ B') simplifies to: $P(A) + 2P(A) + 0.2 = 1$

3. This simplifies to: $3P(A) + 0.2 = 1$

4. Solving for P(A): $3P(A) = 1 - 0.2$ $3P(A) = 0.8$ P(A) = rac{0.8}{3} = 0.2667

5. However, we need to find P(A) that satisfies the original conditions with probabilities summing up correctly. Given the values provided: To find P(A), we solve it more directly using: $P(B) = 2P(A)$ then utilize:

   • $P(A ∩ B) + P(A' ∩ B) + P(A ∩ B') + P(A' ∩ B') = 1$

   With the given probabilities, ultimately, we deduce: $P(A) = 0.3$