The Venn Diagram represents a group of children who swam (left circle) and built sandcastles (right circle) at the beach.Match the symbol or description on the left with its corresponding value on the right. S stands for the event “Swam” and C stands for the event “built sandcastles.” Assume the numbers represent the entire universe.1. P(S) 0.76 2. P(S or C, but not both) 0.84 3. P(C) 0.60 4. P(S ∪ C) 1.00 5. P(C, but not S) 0.40 6. P(S ∩ C) 0.16

The sample space, is the set of all children involved. Thus,

                   n(Samples space)=n(S)=6+15+4=25.

The letters S and C represent the sets "Swam" and "built castles", clearly.

Let
[tex]E_1[/tex], [tex]E_2[/tex], [tex]E_3[/tex], [tex]E_4[/tex], [tex]E_5[/tex],[tex]E_6[/tex] 
be the following events, with the following number of sets:

[tex]E_1[/tex]: S.        n([tex]E_1[/tex])= number of all those who swam =21.

[tex]E_2[/tex]: S or C, but not both.        n([tex]E_2[/tex])=6+4=10, because there are 6 of those who swam but not built castles, and 4 of those who built castles but not swam.

[tex]E_3[/tex]: C.         n([tex]E_3[/tex])=15+4=19.

[tex]E_4[/tex]: S ∪ C.         n([tex]E_4[/tex])=6+15+4=25.

[tex]E_5[/tex]: C, but not S.         n([tex]E_5[/tex])=4, as there are only 4 who built castles but not swam.

[tex]E_6[/tex]: S ∩ C.         n([tex]E_6[/tex])=15, as they are 15 who built castles and swam.


The formula for the probability [tex]P(E)[/tex] of an event [tex]E[/tex] is [tex]P(E)=n(E)/n(S)[/tex].

Thus, the probabilities are as follows:

[tex]P(E_1)=n(E_1)/n(S)=21/25=0.84[/tex]

[tex]P(E_2)=n(E_2)/n(S)=10/25=0.4[/tex]

[tex]P(E_3)=n(E_3)/n(S)=19/25=0.76[/tex]

[tex]P(E_4)=n(E_4)/n(S)=25/25=1.0[/tex]

[tex]P(E_5)=n(E_5)/n(S)=4/25=0.16[/tex]

[tex]P(E_6)=n(E_6)/n(S)=15/25=0.6[/tex]