describing conditional probabilities\nexplain why $p(a|d)$ and $p(d|a)$ from the table below are not…

Explanation:

Step1: Recall conditional - probability formula

The formula for conditional probability is $P(X|Y)=\frac{P(X\cap Y)}{P(Y)}$. For $P(A|D)$, we have $P(A|D)=\frac{P(A\cap D)}{P(D)}$. From the table, $P(A\cap D)=\frac{2}{17}$ and $P(D)=\frac{10}{17}$, so $P(A|D)=\frac{\frac{2}{17}}{\frac{10}{17}}=\frac{2}{10} = 0.2$.

Step2: Calculate $P(D|A)$

For $P(D|A)$, using the formula $P(D|A)=\frac{P(D\cap A)}{P(A)}$. From the table, $P(D\cap A)=\frac{2}{17}$ and $P(A)=\frac{8}{17}$, so $P(D|A)=\frac{\frac{2}{17}}{\frac{8}{17}}=\frac{2}{8}=0.25$.

Step3: Compare the two probabilities

Since $P(A|D) = 0.2$ and $P(D|A)=0.25$, $P(A|D)\neq P(D|A)$ because the denominators in the conditional - probability formulas (the probabilities of the conditioning events $D$ and $A$ respectively) are different.

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. When calculating $P(A|D)=\frac{P(A\cap D)}{P(D)}$ and $P(D|A)=\frac{P(D\cap A)}{P(A)}$, with $P(A\cap D) = P(D\cap A)$, the different values of $P(D)$ and $P(A)$ lead to different results for the two conditional probabilities.