Question 1184502: Consider the same 4 students as in the previous problem, but suppose that the teacher wants to select 2 students.
(a) Define the set of basic outcomes for this problem (you can abbreviate the names to write less). What is the probability of each outcome?
(b) Let A = “the name of at least one of the selected student starts with A”, B = “the name of at least one of the selected student endswith a”. Find the probabilities of these events.
(c) Find the probabilities of the events A∪B, A∩B.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Let's revisit the problem of selecting 2 students from the group of 4 (Anna, Maria, Alex, Ivan) and calculate the probabilities.
**(a) Basic Outcomes and Probabilities:**
Here are the possible pairs of students the teacher can select (order doesn't matter):
* AM (Anna, Maria)
* AX (Anna, Alex)
* AI (Anna, Ivan)
* MX (Maria, Alex)
* MI (Maria, Ivan)
* XI (Alex, Ivan)
There are 6 possible outcomes. Since the selection is random, each outcome has an equal probability of 1/6.
**(b) Events A and B:**
* **Event A ("at least one starts with A"):** This event includes the outcomes where at least one of the selected students' names starts with "A". The outcomes are:
* AM
* AX
* AI
Therefore, P(A) = 3/6 = 1/2 = 0.5
* **Event B ("at least one ends with a"):** This event includes the outcomes where at least one of the selected students' names ends with "a". The outcomes are:
* AM
* MX
Therefore, P(B) = 2/6 = 1/3 ≈ 0.333
**(c) Combined Probabilities:**
* **A ∪ B (A or B or both):** This is the event that at least one student's name starts with "A" *or* at least one student's name ends with "a" (or both). The outcomes are:
* AM
* AX
* AI
* MX
Therefore, P(A ∪ B) = 4/6 = 2/3 ≈ 0.667
* **A ∩ B (A and B):** This is the event that at least one student's name starts with "A" *and* at least one student's name ends with "a". The only outcome that satisfies both conditions is:
* AM
Therefore, P(A ∩ B) = 1/6 ≈ 0.167
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