Given that P(A) = 0.35 , P(B) = 0.45 and P(A ∩ B) = 0.13 find a) P(A ∪ B) b) P(A' | B') The event C has P(C) = 0.20 - Edexcel - A-Level Maths Statistics - Question 7 - 2013 - Paper 1

To find the conditional probability, we use:

$P(A' | B') = \frac{P(A' ∩ B')}{P(B')}$

First, we need to find $P(A' ∩ B')$:

Using the complement rule:

$P(A' ∩ B') = 1 - P(A ∪ B)$

Substituting the known value:

$P(A' ∩ B') = 1 - 0.67 = 0.33$

Now substituting into the conditional probability formula, where $P(B') = 1 - P(B) = 0.55$:

$P(A' | B') = \frac{0.33}{0.55} \approx 0.6$

So, the answer is:

P(A' | B') ≈ 0.6

Since B and C are independent events, the formula for the intersection is:

$P(B ∩ C) = P(B) imes P(C)$

Substituting the given values:

$P(B ∩ C) = 0.45 imes 0.20 = 0.09$

Thus, the answer is:

P(B ∩ C) = 0.09

To draw the Venn diagram, we represent the three events A, B, and C as overlapping circles. The probabilities for each region can be computed as follows:

1. P(A) = 0.35
2. P(B) = 0.45
3. P(C) = 0.20
4. P(A ∩ B) = 0.13, hence, the intersection of A and B should show 0.13.
5. Mutually exclusive region for C should be illustrated, which means it does not overlap with A. The remaining probabilities can be calculated based on the total area being 1 with overlaps accounted for. It's essential to also denote $P(B ∩ C) = 0.09$ from part (c).

The diagram should clearly label each region with the corresponding probabilities.

To find the probability of the union of B and the complement of C, we use:

$P(B ∪ C') = P(B) + P(C') - P(B ∩ C')$

First, calculate $P(C')$:

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

Next, we calculate $P(B ∩ C')$. Knowing B and C are independent:

$P(B ∩ C') = P(B) imes P(C') = 0.45 imes 0.80 = 0.36$

Now substitute these into the union formula:

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

Thus, the answer is:

P(B ∪ C') = 0.89