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Probability of Multiple Events
The Venn Diagram as a Tool



Venn diagrams can be used as a tool to help breakdown
problems of multiple events into much simpler problems.  For instance:
VennAandB


The probability of A and B can be by this Venn.  This help you to see that to get this we will need to find only what things belong to both.  To do this, we multiply the probability of A by the of B.  Mathematically it looks like this:


                                    P(A and B) = P(A)P(B)











VennAor         

           
The probability of A or B can be represented by this Venn.  This helps you to see that to get this probability we will need to take away the parts that the two events share.  So A and B needs to be added together and the we need to take away the part that A and B shares. Mathematically this means:
 
                             P(A or B)=P(A)+P(B)-P(A and B)






If  we combine the two basic ideas illustrated above with the basic definition of probability: If  we combine the two basic ideas illustrated above with the basic
# of times a specific event occurs
   # of total events in the group

we can tackle more complicated probabilities with even more events.


Try to solve the following probability problem using the Venn diagram worksheet provided.
 

Venn Worksheet

Problem:

A high school is offering three languages:  one in Spanish, one in French, and one in German.  These classes are open to any of the 100 students in the school.  There are 28 students in the Spanish class, 26 in the French class, and 16 in the German class.  There are 12 students that are in both Spanish and French, 4 that are in both Spanish and German, and 6 that are in both French and German.  In addition there are 2 students taking all three classes.

(a)  If a student is chosen randomly, what is the
that he or she is taking exactly one language class?
(b) If a student is chosen randomly, what is the
that he or she is taking exactly one language class?



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