SOLUTION: A survey of 100 students asked if they studied a foreign language. The result showed: Spanish, 28; German, 30; French, 42; Spanish and German, 8; Spanish and French,10; German and

Algebra.Com
Question 1048656: A survey of 100 students asked if they studied a foreign language. The result showed: Spanish, 28; German, 30; French, 42; Spanish and German, 8; Spanish and French,10; German and French 5; all three languages,3. What is the probability that a randomly selected student studied no foreign language?
Found 2 solutions by rothauserc, ikleyn:
Answer by rothauserc(4718)   (Show Source): You can put this solution on YOUR website!
28 + 30 + 42 = 100
:
p = 1, q = 0, n = 100, r = 0
:
P ( r = 0 ) = 100C0 * 1^0 * 0^100 = 1 * 1 * 0 = 0
:
**********************************************
Probability that a randomly selected student studies no
foreign language is 0
**********************************************
:
Answer by ikleyn(52790)   (Show Source): You can put this solution on YOUR website!
.
A survey of 100 students asked if they studied a foreign language. The result showed:
Spanish, 28; German, 30; French, 42; Spanish and German, 8; Spanish and French,10; German and French 5; all three languages,3.
What is the probability that a randomly selected student studied no foreign language?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

Let S be the subset of the 100 students studying Spanish, and let |S| be the cardinality of the subset S. We are given |S| = 28.

Let G be the subset of the 100 students studying German, and let |G| be the cardinality of the subset G. We are given |G| = 30.

Let F be the subset of the 100 students studying French, and let |F| be the cardinality of the subset F. We are given |F| = 42.

Let SG be the subset of the 100 students studying Spanish and German. 
    It is the intersection of the sets S and G. 
    And let |SG| be the cardinality of the subset SG. We are given |SG| = 8.

Let SF be the subset of the 100 students studying Spanish and French. 
    It is the intersection of the sets S and F. 
    And let |SF| be the cardinality of the subset SF. We are given |SF| = 10.

Let GF be the subset of the 100 students studying German and French. 
    It is the intersection of the sets G and F. 
    And let |GF| be the cardinality of the subset GF. We are given |GF| = 5.

Finally, let SGF be the subset of the 100 students studying Spanish, German and French. 
    It is the intersection of the sets S, G and F. 
    And let |SGF| be the cardinality of the subset SGF. We are given |GF| = 3.


Now, the number of students among of 100 surveyed who studies at least one of these three languages is

   N = |S| + |G| +|F| - |SG| - |SF| - |GF| + |SGF| = 28 + 30 + 42 - 8 - 10 - 5 + 3 = 80.

Thus 80 students of 100 learn at least one language.
The rest of 100, 100-80 = 20, do not study these languages.


Therefore, the probability that a randomly selected student studied no foreign language is  = .


Answer.  The probability that a randomly selected student studied no foreign language is .

The formula we used is well known in the elementary set theory.
Its proof is very straightforward and simple.
See, for example, the lesson
    - Advanced problems on counting elements in sub-sets of a given finite set
in this site.

Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lesson is the part of this online textbook in the topic "Miscellaneous word problems" of the section "Word problems".


RELATED QUESTIONS

There are a total of 108 foreign language students in a high school where they offer... (answered by edjones)
Each student at Sussex Elementary School takes one foreign language. Two thirds of the... (answered by josmiceli)
The Department of Foreign Languages of a liberal arts college conducted a survey of its... (answered by ikleyn)
The Department of Foreign Languages of a liberal arts college conducted a survey of its... (answered by rfer)
In a survey of 100 student the number of student studying various language where find to... (answered by Theo,jim_thompson5910)
I am given a pie chart with three foreign languages on it, French( 25%), Italian (20%),... (answered by davethejackal)
suppose that 100 of the 120 mathematics students at a college take at least one of the... (answered by ikleyn)
An elementary school is offering 3 language classes: one in Spanish, one in French, and... (answered by ikleyn)
At Madison High School each student studies exactly one foreign language. Three-fifths of (answered by Theo)