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 93 students in the school. There are 35 in the Spanish class, 31 in the French class, and 22 in the German class. There are 12 students that in both Spanish and French, 6 are in both Spanish and German, and 8 are in both French and German. In addition, there are 2 students taking all 3 classes.
If one student is chosen randomly, what is the probability that he or she is not in any of these classes?
If two students are chosen randomly, what is the probability that at least one of them is taking a language class?
Show work or an explanation please!
draw a Venn diagram of three intersecting circles enclosed in a box
label the circles S, F, and G
S/F/G -- 2 ... these are in all the two way intersections
S/F -- 10 + (2)
S/G -- 4 + (2)
F/G -- 6 + (2)
S -- 19 + (2) + (10) + (4)
F -- 13 + (2) + (10) + (6)
G -- 10 + (2) + (4) + (6)
language students
... 19 + 13 + 10 + 6 + 4 + 10 + 2 = 64
non-language ... 93 - 64
"at least one" means that both cannot be non-language
To find the probability that a randomly chosen student is not in any of the language classes, we need to calculate the number of students who are not in any of the classes and divide it by the total number of students.
There are 93 students in the school, and we know the number of students in each language class:
35 in Spanish class
31 in French class
22 in German class
To find the number of students who are not in any of the classes, we subtract the total number of students in the language classes from the total number of students in the school:
93 - (35 + 31 + 22) = 93 - 88 = 5
Therefore, there are 5 students who are not in any of the language classes.
To calculate the probability, we divide the number of students who are not in any of the language classes by the total number of students:
Probability = Number of students not in any language class / Total number of students
Probability = 5 / 93 = 0.0538 (rounded to 4 decimal places)
The probability that a randomly chosen student is not in any of the language classes is approximately 0.0538.
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To find the probability that at least one of the two randomly chosen students is taking a language class, we need to calculate the probability that both students are not taking any language classes and subtract it from 1.
The probability that a randomly chosen student is not taking any language class was calculated earlier as 0.0538.
For both randomly chosen students to not be taking any language classes, we need to multiply their individual probabilities of not taking a language class:
Probability that 1st student is not taking a language class = 5 / 93
Probability that 2nd student is not taking a language class = 4 / 92 (after removing the 1 student who has already been chosen)
To calculate the probability that both students are not taking any language classes, we multiply these two probabilities together:
Probability that both students are not taking any language class = (5 / 93) * (4 / 92) = 0.0221 (rounded to 4 decimal places)
Finally, to find the probability that at least one of the two randomly chosen students is taking a language class, we subtract the probability that both students are not taking any language classes from 1:
Probability = 1 - 0.0221 = 0.9779 (rounded to 4 decimal places)
The probability that at least one of the two randomly chosen students is taking a language class is approximately 0.9779.
To find the probability that a randomly chosen student is not in any of these classes, we need to determine the number of students who are not taking any language class.
We can start by finding the total number of students who are taking at least one language class. To do this, we add up the number of students in each individual language class (35 + 31 + 22 = 88). However, this count includes the students who are taking multiple language classes multiple times.
To eliminate the duplicates, we need to subtract the number of students who are taking multiple language classes. We do this by subtracting the number of students taking two classes, twice the number of students taking all three classes, from our count.
The number of students taking two classes is calculated by adding together the number of students in each combination of two classes: (12 + 6 + 8 = 26).
The number of students taking all three classes is given as 2.
Therefore, the total number of students who are taking at least one language class is (88 - 26 - 2 = 60).
Now, to find the probability that a randomly chosen student is not in any of the language classes, we subtract the number of students in the language classes from the total number of students in the school.
Total number of students in the school: 93
Number of students in language classes: 60
Therefore, the number of students not in any of the language classes is (93 - 60 = 33).
The probability that a randomly chosen student is not in any of these classes is given by dividing the number of students not in the language classes by the total number of students:
Probability = Number of students not in language classes / Total number of students
Probability = 33 / 93
Probability ≈ 0.3548
Thus, the probability that a randomly chosen student is not in any of the language classes is approximately 0.3548, or 35.48%.
To find the probability that at least one of two randomly chosen students is taking a language class, we will use the complement rule.
The complement rule states that the probability of an event occurring is equal to 1 minus the probability of the event not occurring.
First, let's find the probability that neither of the two students is taking a language class.
Since we already know the probability that a randomly chosen student is not in any of the language classes (0.3548 or 35.48%), the probability that neither of the two students is taking a language class is:
Probability that neither student is taking a language class = 0.3548^2 = 0.1258 (approximately).
Now, we can find the probability that at least one of the two students is taking a language class by subtracting the probability that neither student is taking a language class from 1:
Probability that at least one student is taking a language class = 1 - 0.1258 = 0.8742 (approximately).
Therefore, the probability that at least one of two randomly chosen students is taking a language class is approximately 0.8742, or 87.42%.