explain why $p(a|d)$ and $p(d|a)$ from the table below are not equal.\n| | c | d | total |\n|----|----|----|…

Explanation:

Step1: Recall conditional - probability formula

The formula for conditional probability is $P(X|Y)=\frac{P(X\cap Y)}{P(Y)}$.

Step2: Calculate $P(A|D)$

$P(A|D)=\frac{P(A\cap D)}{P(D)}$. From the table, $A\cap D$ has 2 elements, $P(A\cap D)=\frac{2}{17}$, and $P(D)=\frac{10}{17}$. So $P(A|D)=\frac{\frac{2}{17}}{\frac{10}{17}}=\frac{2}{10} = 0.2$.

Step3: Calculate $P(D|A)$

$P(D|A)=\frac{P(D\cap A)}{P(A)}$. From the table, $D\cap A$ has 2 elements, $P(D\cap A)=\frac{2}{17}$, and $P(A)=\frac{8}{17}$. So $P(D|A)=\frac{\frac{2}{17}}{\frac{8}{17}}=\frac{2}{8}=0.25$.

Step4: Compare the two probabilities

Since $P(A|D) = 0.2$ and $P(D|A)=0.25$, $P(A|D)\neq P(D|A)$. The reason is that in the conditional - probability formula, the denominators $P(D)$ and $P(A)$ are different, which changes the value of the fraction even though the numerator $P(A\cap D)=P(D\cap A)$ in both cases.

Answer:

$P(A|D)$ and $P(D|A)$ are not equal because in the conditional - probability formula $P(X|Y)=\frac{P(X\cap Y)}{P(Y)}$, the denominators $P(D)$ and $P(A)$ are different, resulting in different values for the two conditional probabilities. Specifically, $P(A|D) = 0.2$ and $P(D|A)=0.25$.