A sample space contains 7 sample points and events A and C as seen in the Venn diagram. (The ven diagram has 1,5,6 in event A (circle 1) 2,3 in overlap/intersection and 4,7 in second circle or event B)

Let P(1)=P(2)=P(3)=P(7)=0.1

P(4)=P(5)=0.05

and P(6)=0.5

Calculate P(A^C)

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.

A sample space contains 7 sample points and events A and C as seen in the Venn diagram. Let P(1)=P(2)=P(3)=P(7)=0.1, P(4)=P(5)=0.05, and P(6)=0.5. Calculate P(A^C).

Calculate the probability of the intersection of events A and C using a Venn diagram with sample points. Learn how to apply the addition rule of probability to find P(A ∩ C) based on the given probabilities for sample points. Suitable for Grades 10-12.

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