Question 1184501
Let's break down this probability problem with Venn diagrams.

**(a) Basic Outcomes and Probabilities:**

*   **Set of Basic Outcomes:** {Anna, Maria, Alex, Ivan}
*   **Probability of each outcome:** Since the selection is random, each student has an equal chance of being chosen. Therefore, the probability of each outcome is 1/4 or 0.25.

**(b) Events A and B:**

*   **Event A ("starts with A"):** {Anna, Alex}  P(A) = 2/4 = 1/2 = 0.5
*   **Event B ("ends with a"):** {Anna, Maria}  P(B) = 2/4 = 1/2 = 0.5

**(c) Combined Probabilities and Venn Diagrams:**

Here's a breakdown of each combined probability along with a description of the corresponding Venn diagram.  Imagine two overlapping circles. One represents event A (starts with A), and the other represents event B (ends with a).

1.  **A ∪ B (A or B or both):**

    *   **Outcomes:** {Anna, Maria, Alex}
    *   **Probability:** P(A ∪ B) = 3/4 = 0.75
    *   **Venn Diagram:** Shade *both* circles and the overlapping area in the center. This represents all outcomes in A, all outcomes in B, and any outcomes that are in both.

2.  **A ∩ B (A and B):**

    *   **Outcomes:** {Anna}
    *   **Probability:** P(A ∩ B) = 1/4 = 0.25
    *   **Venn Diagram:** Shade *only* the overlapping area in the center of the two circles. This represents the outcomes that are in *both* A and B.

3.  **A \ B (A but not B):**

    *   **Outcomes:** {Alex}
    *   **Probability:** P(A \ B) = 1/4 = 0.25
    *   **Venn Diagram:** Shade the part of circle A that *does not* overlap with circle B. This represents outcomes that are in A but *not* in B.

4.  **(not A) ∪ (not B) (Not A or Not B):**

    *   **Outcomes:** {Maria, Ivan, Alex}
    *   **Probability:** P((not A) ∪ (not B)) = 3/4 = 0.75  (This is the same as not(A and B), by De Morgan's Law)
    *   **Venn Diagram:** Shade everything *outside* the area where the two circles overlap.  This represents outcomes that are *not* in A, outcomes that are *not* in B, and all outcomes that are not in the intersection.

5.  **(not A) ∩ (not B) (Not A and Not B):**

    *   **Outcomes:** {Ivan}
    *   **Probability:** P((not A) ∩ (not B)) = 1/4 = 0.25 (Only Ivan satisfies neither condition.)
    *   **Venn Diagram:** Shade the area *outside* of both circles. This represents the outcomes that are *neither* in A *nor* in B.