In this problem, we are tasked with calculating $P(A \cap C)$, which represents the probability of the intersection between events $A$ and $C$, based on the information provided in the Venn diagram.

Step 1: Identify the sample points in $A \cap C$

From the Venn diagram description:

• $A$ contains sample points 1, 5, 6.
• $C$ contains sample points 2, 3, 4, 7.
• The overlap (intersection) between $A$ and $C$ includes sample points 2 and 3.

Thus, $A \cap C$ contains sample points 2 and 3.

Step 2: Sum the probabilities of the sample points in $A \cap C$

We are given the probabilities for the sample points:

• $P(2) = 0.1$
• $P(3) = 0.1$

The total probability of $A \cap C$ is the sum of these probabilities: $P(A \cap C) = P(2) + P(3) = 0.1 + 0.1 = 0.2$

Thus, $P(A \cap C) = 0.2$.

Would you like further details or have any additional questions?

Related Questions:

1. How do you calculate $P(A \cup C)$?
2. What is the complement of $A \cap C$?
3. How would the probability change if a sample point had different weights?
4. How do you interpret disjoint events in probability?
5. What is the total probability of all the sample points in this space?

Tip:

In probability problems involving Venn diagrams, intersections represent simultaneous occurrences, and their probabilities can be calculated by summing the relevant sample points in that region.