Among 40 students: $$n(A)=20,\quad n(B)=18,\quad n(C)=16,$$\[n(A\cap B)=8,\quad n(A\cap C)=7,\quad n

Question

Among 40 students: $$n(A)=20,\quad n(B)=18,\quad n(C)=16,$$$$n(A\cap B)=8,\quad n(A\cap C)=7,\quad n(B\cap C)=6,$$ and $n(A\cap B\cap C)=3$. Draw a three-set Venn diagram and fill all eight regions, including “none”. Find the probability that a randomly selected student belongs to exactly two sets. Find the probability that the student belongs to none.

Accepted Answer

(a) $A$ only: $8$; $B$ only: $7$; $C$ only: $6$; $A\cap B$ only: $5$; $A\cap C$ only: $4$; $B\cap C$ only: $3$; all three: $3$; none: $4$. (b) $\frac3{10}$. (c) $\frac1{10}$.

Probability

Probability

Solution:

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