Question

use the venn diagram to calculate probabilities. which probability is correct? p(a) = 3/5 p(b) = 16/31 p(a|b) = 2/7 p(b|a) = 10/21

Explanation:

Step1: Calculate the total number of elements in the universal set

$15 + 6+10 + 4=35$

Step2: Calculate $P(A)$

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

Step3: Calculate $P(B)$

$n(B)=6 + 10=16$, so $P(B)=\frac{16}{35}$

Step4: Calculate $P(A|B)$

By the formula $P(A|B)=\frac{P(A\cap B)}{P(B)}$, $P(A\cap B)=\frac{6}{35}$, $P(B)=\frac{16}{35}$, so $P(A|B)=\frac{6}{16}=\frac{3}{8}$

Step5: Calculate $P(B|A)$

By the formula $P(B|A)=\frac{P(A\cap B)}{P(A)}$, $P(A\cap B)=\frac{6}{35}$, $P(A)=\frac{21}{35}$, so $P(B|A)=\frac{6}{21}=\frac{2}{7}$

Answer:

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