Question 1184503
Let's break down these probability problems step by step.

**Problem 1: Selecting 1 Student**

**(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:**

* **A ∪ B (A or B or both):** {Anna, Maria, Alex}  P(A ∪ B) = 3/4 = 0.75  (This is because 3 out of the 4 students satisfy either condition A or B or both.)
* **A ∩ B (A and B):** {Anna}  P(A ∩ B) = 1/4 = 0.25 (Only Anna satisfies both conditions.)
* **A \ B (A but not B):** {Alex}  P(A \ B) = 1/4 = 0.25 (Only Alex starts with A but doesn't end with a.)
* **(not A) ∪ (not B) (Not A or Not B):** {Ivan, Maria, Alex} P((not A) ∪ (not B)) = 3/4 = 0.75.  (This is the same as not(A and B), by De Morgan's Law)
* **(not A) ∩ (not B) (Not A and Not B):** {Ivan}  P((not A) ∩ (not B)) = 1/4 = 0.25 (Only Ivan satisfies neither condition.)

**Problem 2: Selecting 2 Students**

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

We'll use abbreviations: A = Anna, M = Maria, X = Alex, I = Ivan.  Since we're selecting 2 students, the order doesn't matter (combinations, not permutations). The possible pairs are:

* {A, M}
* {A, X}
* {A, I}
* {M, X}
* {M, I}
* {X, I}

There are 6 possible outcomes.  Since the selection is random, each outcome has a probability of 1/6.

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

* **Event A ("at least one starts with A"):** {A, M}, {A, X}, {A, I}  P(A) = 3/6 = 1/2 = 0.5
* **Event B ("at least one ends with a"):** {A, M}, {M, X}  P(B) = 2/6 = 1/3 ≈ 0.333

**(c) Combined Probabilities:**

* **A ∪ B (at least one starts with A *or* at least one ends with a):**  {A, M}, {A, X}, {A, I}, {M, X}. P(A ∪ B) = 4/6 = 2/3 ≈ 0.667
* **A ∩ B (at least one starts with A *and* at least one ends with a):** {A, M} P(A ∩ B) = 1/6 ≈ 0.167