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

explain why p(a|d) and p(d|a) from the table below are not equal.\n| | c | d | total |\n|--|--|--|--|\n| a | 6 | 2 | 8 |\n| b | 1 | 8 | 9 |\n| total | 7 | 10 | 17 |

explain why p(a|d) and p(d|a) from the table below are not equal.\n| | c | d | total |\n|--|--|--|--|\n| a | 6 | 2 | 8 |\n| b | 1 | 8 | 9 |\n| total | 7 | 10 | 17 |

Answer

Explanation:

Step1: Recall conditional - probability formula

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

Step2: Calculate $P(A|D)$

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

Step3: Calculate $P(D|A)$

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

Step4: Compare the two probabilities

Since $0.2\neq0.25$, $P(A|D)$ and $P(D|A)$ are not equal.

Answer:

$P(A|D)$ and $P(D|A)$ are not equal because when calculated using the conditional - probability formula $P(X|Y)=\frac{P(X\cap Y)}{P(Y)}$, $P(A|D) = 0.2$ and $P(D|A)=0.25$.