An elementary school is offering 3 language classes: one in Spanish, one in French, and one in German. These classes are open to any of the 107 students in the school. There are 42 in the Spanish class, 32 in the French class, and 23 in the German class. There are 14 students that in both Spanish and French, 8 are in both Spanish and German, and 9 are in both French and German. In addition, there are 4 students taking all 3 classes.

If one student is chosen randomly, what is the probability that he or she is taking at least two language classes?
If two students are chosen randomly, what is the probability that at least one of them is taking German?

Make a Venn diagram, and you can easily see that there are 70 students taking at least one of the classes, and thus 37 taking no class.

Using the diagram, you can easily see how many students fit the questions; and thus the probabilities needed.

To find the probability that a randomly chosen student is taking at least two language classes, we need to determine the number of students taking two or three language classes.

To get this number, we can use the principle of inclusion-exclusion. We start by adding the number of students in each language class:

Students in Spanish class = 42
Students in French class = 32
Students in German class = 23

However, this counts some students multiple times because there are students taking multiple language classes. We need to subtract the students who are taking two language classes:

Students in both Spanish and French = 14
Students in both Spanish and German = 8
Students in both French and German = 9

But we have subtracted some students twice, so we need to add back those who are taking all three language classes:

Students taking all three classes = 4

Now, we can calculate the total number of students taking at least two language classes:

Total students taking at least two language classes = (Students in Spanish class + Students in French class + Students in German class) - (Students in both Spanish and French + Students in both Spanish and German + Students in both French and German) + Students taking all three classes
= (42 + 32 + 23) - (14 + 8 + 9) + 4
= 75

The probability that a randomly chosen student is taking at least two language classes can be calculated by dividing the total number of students taking at least two language classes by the total number of students in the school:

P(student taking at least two language classes) = Total students taking at least two language classes / Total number of students
= 75 / 107
= 0.7009 (rounded to four decimal places)

Therefore, the probability that a randomly chosen student is taking at least two language classes is approximately 0.7009.

Now let's move on to the second question regarding the probability that at least one of two randomly chosen students is taking German.

To calculate this probability, we need to find the probability that both students are not taking German and subtract it from 1 (since 1 minus the probability of an event not happening gives the probability of the event happening).

The probability that the first student is not taking German can be calculated by subtracting the number of students taking German from the total number of students:

P(first student not taking German) = (Total number of students - Students taking German) / Total number of students
= (107 - 23) / 107
= 84 / 107
= 0.785 (rounded to three decimal places)

Since the choices are made randomly, the probability that the second student is not taking German is the same as the probability for the first student:

P(second student not taking German) = P(first student not taking German) = 0.785

Therefore, the probability that both students are not taking German is obtained by multiplying their individual probabilities:

P(both students not taking German) = P(first student not taking German) * P(second student not taking German)
= 0.785 * 0.785
= 0.615 (rounded to three decimal places)

Finally, we can calculate the probability that at least one of the two students is taking German by subtracting the probability that neither student is taking German from 1:

P(at least one of two students taking German) = 1 - P(both students not taking German)
= 1 - 0.615
= 0.385 (rounded to three decimal places)

Therefore, the probability that at least one of the two randomly chosen students is taking German is approximately 0.385.

To find the probability that a randomly chosen student is taking at least two language classes, we need to calculate the number of students taking exactly two classes and the number of students taking all three classes.

Let's denote:
A = number of students taking Spanish
B = number of students taking French
C = number of students taking German
n(A) = number of students taking Spanish only
n(B) = number of students taking French only
n(C) = number of students taking German only
n(AB) = number of students taking both Spanish and French
n(AC) = number of students taking both Spanish and German
n(BC) = number of students taking both French and German
n(ABC) = number of students taking all three classes

From the given information:
A = 42
B = 32
C = 23
n(AB) = 14
n(AC) = 8
n(BC) = 9
n(ABC) = 4

To find the number of students taking exactly two classes, we sum the students taking exactly two classes in each overlap:

n(AB) + n(AC) + n(BC) - 2n(ABC) = 14 + 8 + 9 - 2(4) = 31

Now, to find the probability that a randomly chosen student is taking at least two language classes, we divide the total number of students taking at least two classes by the total number of students in the school:

P(taking at least two classes) = (31 + 4) / 107 = 35 / 107 ≈ 0.327

Therefore, the probability that a randomly chosen student is taking at least two language classes is approximately 0.327.

To find the probability that at least one of two randomly chosen students is taking German, we need to consider the complementary event and then subtract it from 1.

The probability that neither of the two students is taking German is:
(1 - probability of choosing a German-speaking student on the first pick) ×
(1 - probability of choosing a German-speaking student on the second pick)

Probability of not choosing a German-speaking student on the first pick:
= (number of non-German students) / (total number of students)
= (107 - 23) / 107
= 84 / 107

Probability of not choosing a German-speaking student on the second pick:
= (number of non-German students after the first pick) / (total number of students after the first pick)
= (107 - 23 - 1) / (107 - 1)
= 83 / 106

Thus, the probability that neither student is taking German is:
(84 / 107) × (83 / 106)

To find the probability that at least one of the two students is taking German, we subtract the probability from 1:

P(at least one student taking German) = 1 - (84 / 107) × (83 / 106)
= 1 - 6972 / 11242
= 4270 / 11242
≈ 0.379

Therefore, the probability that at least one of two randomly chosen students is taking German is approximately 0.379.