Question 1048656
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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?
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<pre>
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 {{{20/100}}} = {{{1/5}}}.


<U>Answer</U>.  The probability that a randomly selected student studied no foreign language is {{{1/5}}}.
</pre>

The formula we used is well known in the elementary set theory.

Its proof is very straightforward and simple.

See, for example, the lesson 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/misc/Advanced-probs-counting-elements-in-sub-sets-of-a-given-finite-set.lesson>Advanced problems on counting elements in sub-sets of a given finite set</A>

in this site.


Also, you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


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