use the venn diagram to calculate probabilities. which probability is correct? o $p(a)=\frac{3}{5}$ o…

Explanation:

Step1: Calculate total number of elements

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:

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