use the venn diagram to calculate conditional probabilities. which conditional probabilities are correct…

use the venn diagram to calculate conditional probabilities. which conditional probabilities are correct? check all that apply. $p(d|f)=\frac{6}{34}$ $p(e|d)=\frac{7}{25}$ $p(d|e)=\frac{7}{25}$ $p(f|e)=\frac{8}{18}$ $p(e|f)=\frac{13}{21}$

use the venn diagram to calculate conditional probabilities. which conditional probabilities are correct? check all that apply. $p(d|f)=\frac{6}{34}$ $p(e|d)=\frac{7}{25}$ $p(d|e)=\frac{7}{25}$ $p(f|e)=\frac{8}{18}$ $p(e|f)=\frac{13}{21}$

Answer

Explanation:

Step1: Recall conditional - probability formula

The formula for conditional probability is $P(A|B)=\frac{P(A\cap B)}{P(B)}=\frac{n(A\cap B)}{n(B)}$, where $n(A\cap B)$ is the number of elements in the intersection of $A$ and $B$, and $n(B)$ is the number of elements in $B$.

Step2: Calculate $P(D|F)$

$n(D\cap F)=1 + 5=6$, $n(F)=1+7 + 21+5=34$. So $P(D|F)=\frac{n(D\cap F)}{n(F)}=\frac{6}{34}$.

Step3: Calculate $P(E|D)$

$n(E\cap D)=6 + 5=11$, $n(D)=13+6 + 5+1=25$. So $P(E|D)=\frac{n(E\cap D)}{n(D)}=\frac{11}{25}\neq\frac{7}{25}$.

Step4: Calculate $P(D|E)$

$n(D\cap E)=6 + 5=11$, $n(E)=4+6 + 1+7=18$. So $P(D|E)=\frac{n(D\cap E)}{n(E)}=\frac{11}{18}\neq\frac{7}{25}$.

Step5: Calculate $P(F|E)$

$n(F\cap E)=1 + 7=8$, $n(E)=4+6 + 1+7=18$. So $P(F|E)=\frac{n(F\cap E)}{n(E)}=\frac{8}{18}$.

Step6: Calculate $P(E|F)$

$n(E\cap F)=1 + 7=8$, $n(F)=1+7 + 21+5=34$. So $P(E|F)=\frac{n(E\cap F)}{n(F)}=\frac{8}{34}\neq\frac{13}{21}$.

Answer:

$P(D|F)=\frac{6}{34}$, $P(F|E)=\frac{8}{18}$