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 $P(A' | B')$, we can use the formula: $P(A' | B') = \frac{P(A' ∩ B')}{P(B')}$

First, we calculate $P(B')$: $P(B') = 1 - P(B) = 1 - 0.45 = 0.55$

Next, we find $P(A' ∩ B')$. Using the complement rule: $P(A' ∩ B') = 1 - P(A ∪ B) = 1 - 0.67 = 0.33$

Now substituting these values: $P(A' | B') = \frac{0.33}{0.55} ≈ \frac{3}{5}$

Therefore, $P(A' | B')$ is approximately 0.6.

Given that B and C are independent events, we can use: $P(B ∩ C) = P(B) imes P(C)$

Substituting the values: $P(B ∩ C) = 0.45 × 0.20 = 0.09$

Therefore, the probability $P(B ∩ C)$ is 0.09.

To draw a Venn diagram, we need to allocate the probabilities appropriately. The diagram should reflect that:

• The area for event A has a probability of 0.35.
• The area for event B has a probability of 0.45.
• The intersection of A and B has a probability of 0.13.
• The area for event C has a probability of 0.20, and it does not overlap with A as they are mutually exclusive.

In the Venn diagram:

• The region exclusive to A would be $0.35 - 0.13 = 0.22$.
• The region exclusive to B would be $0.45 - 0.13 = 0.32$.
• The overlap would remain as $0.13$.
• The intersection area for C with respect to B can be represented as $0.20$ aside from A.

This information can be illustrated graphically.

To find the probability of the complement of the union of B and C, we will use: $P([B ∪ C]') = 1 - P(B ∪ C)$

First, we need to calculate $P(B ∪ C)$: Using the formula: $P(B ∪ C) = P(B) + P(C) - P(B ∩ C)$

Substituting the known values: $P(B ∪ C) = 0.45 + 0.20 - 0.09 = 0.56$

Now, substituting back to find the complement: $P([B ∪ C]') = 1 - 0.56 = 0.44$

Therefore, the probability $P([B ∪ C]')$ is 0.44.