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 Pure - Question 14 - 2021 - Paper 3

To find P(A), we start with the provided information:

1. We know that the total probability of all events sums up to 1:

   $P(A) + P(A') = 1$

2. From the information given:

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

   Hence, we can find P(A ∩ B'):

   $P(A ∩ B') = P(A) - P(A ∩ B)$

   Since P(A ∩ B) = 0.1, we have: $P(A ∩ B') = P(A) - 0.1$

3. To find P(A'), we use the equation:

   $P(A') = 1 - P(A)$

4. We can use the law of total probability:

   $P(B) = P(A ∩ B) + P(A' ∩ B)$

   Given that P(B) = 2P(A), we can substitute:

   $2P(A) = P(A ∩ B) + P(A' ∩ B)$

   Using the previous results, this leads us to:

   $2P(A) = 0.1 + P(A' ∩ B)$

5. We also know:

   $P(A' ∩ B) = P(B) - P(A ∩ B)$

   Thus we can substitute:

   $P(A' ∩ B) = 2P(A) - 0.1$

6. Solving these equations enables us to isolate P(A):

   1. Substitute P(A' ∩ B) into the equation: $2P(A) = 0.1 + (2P(A) - 0.1)$

   2. Upon simplifying, we can establish: $P(A) + P(A') - 0.1 = 1 - 0.2$

7. Solving reveals:

   $P(A) = 0.3$