Question 1184501: A teacher wants to select randomly a student from a group of 4 students. The names of the students are Anna, Maria, Alex, Ivan.
(a) Define the set of basic outcomes for this problem. What is the probability of each outcome?
(b) Consider the random events A = “the name of the selected student starts with A”, B = “the name of the selected student ends with a”. What outcomes do they consist of? What are their probabilities?
(c) Find the probabilities of the events A∪B, A∩B, A\B, (notA)∪(not B), (not A)∩(not B).
Include Venn diagrams(“circles”) for each of 5 cases in(c); shade the areas in question.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! 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.
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