Given that P(A) = 0.35, P(B) = 0.45 and P(A ∩ B) = 0.13 find a) P(A' | B') b) Explain why the events A and B are not independent - Edexcel - A-Level Maths Statistics - Question 4 - 2017 - Paper 2

Two events A and B are independent if:

$P(A \cap B) = P(A) \times P(B)$

Calculating the right side:

$P(A) \times P(B) = 0.35 \times 0.45 = 0.1575$

Since we have:

$P(A \cap B) = 0.13$

And this does not equal 0.1575, we conclude that A and B are not independent.

The Venn diagram should show three overlapping circles labeled A, B, and C. The probabilities for each region should be filled as follows:

• A ∩ B: 0.13 (The overlap between A and B)
• A ∩ C: 0.09 (Mutually exclusive with B)
• B ∩ C: 0.22 (Shared between B and C)
• A ∩ B ∩ C: 0.00 (There is no shared region with A, B, and C)
• Outside all: The remainder percentage derived from the total.

The regions and their probabilities should indicate the contributions of each event accurately.

Using the formula for the union of two events:

$P(B \cup C') = P(B) + P(C') - P(B \cap C')$

First, we calculate P(C'):

$P(C') = 1 - P(C) = 1 - 0.20 = 0.80$

Now we need to determine P(B ∩ C'). Since B and C are independent:

$P(B \cap C') = P(B) \times P(C') = 0.45 \times 0.80 = 0.36$

Now substituting everything back into the union formula:

$P(B \cup C') = 0.45 + 0.80 - 0.36 = 0.89$