For the events A and B, P(A ∩ B') = 0.32, P(A' ∩ B) = 0.11 and P(A ∪ B) = 0.65 - Edexcel - A-Level Maths Statistics - Question 6 - 2006 - Paper 1

To find P(A) and P(B):

1. Calculate P(A):

   • We already know that P(A ∩ B') = 0.32 and from the Venn diagram analysis, we can derive:
   • P(A) = P(A ∩ B') + P(A ∩ B).
2. Calculate P(B):
   • We also know that P(A' ∩ B) = 0.11, and applying the union rule again allows us to derive:
   • P(B) = P(A ∩ B) + P(A' ∩ B).
3. Final values: Based on the above calculations, we find:
   • P(A) = 0.32 + 0.22 = 0.54,
   • P(B) = 0.11 + 0.22 = 0.33.

To determine the conditional probability P(A | B'):

1. Use the formula: $P(A | B') = \frac{P(A \cap B')}{P(B')},$ where P(B') can be found as: $P(B') = 1 - P(B).$
2. Substitute the known values:
   • P(A ∩ B') = 0.32
   • P(B) = 0.33, thus P(B') = 1 - 0.33 = 0.67.
3. Now calculate: $P(A | B') = \frac{0.32}{0.67} \approx 0.478.$

Therefore, P(A | B') is approximately 0.478.

To assess independence, we check:

1. The condition for independence is: $P(A \cap B) = P(A) \cdot P(B).$
2. From previous calculations:
   • P(A ∩ B) is derived from the Venn diagram and can be computed based on the values of P(A) and P(B) calculated earlier.
3. If at any point P(A ∩ B) does not equal P(A) × P(B), then A and B are not independent.
4. Final calculation yields
   • P(A ∩ B) is derived from
   • Comparing values leads us to conclude:
   • Therefore, A and B are NOT independent.