use the venn diagram to calculate probabilities. which probability is correct? p(a) = 3/5 p(b) = 16/31…

Explanation:

Step1: Calculate total elements in the sample - space

The total number of elements in the universal set (U) is (15 + 6+10 + 4=35).

Step2: Calculate (P(A))

(n(A)=15 + 6=21), so (P(A)=\frac{n(A)}{n(U)}=\frac{21}{35}=\frac{3}{5}).

Step3: Calculate (P(B))

(n(B)=6 + 10=16), so (P(B)=\frac{n(B)}{n(U)}=\frac{16}{35}\neq\frac{16}{31}).

Step4: Calculate (P(A|B))

By the formula (P(A|B)=\frac{P(A\cap B)}{P(B)}), and (n(A\cap B) = 6), (n(B)=16), so (P(A|B)=\frac{n(A\cap B)}{n(B)}=\frac{6}{16}=\frac{3}{8}\neq\frac{2}{7}).

Step5: Calculate (P(B|A))

By the formula (P(B|A)=\frac{P(A\cap B)}{P(A)}), and (n(A\cap B) = 6), (n(A)=21), so (P(B|A)=\frac{n(A\cap B)}{n(A)}=\frac{6}{21}=\frac{2}{7}\neq\frac{10}{21}).

Answer:

(P(A)=\frac{3}{5})