Answer:
The Probability that Event A and Event D occur is equal to the probability Event A occurs times the probability that Event D occurs, given that A has occurred.
[tex]P(A\cap D)=P(A)\cdot P(D/A)[/tex]
We can find the values of [tex]P(A/D)[/tex] and [tex]P(D/A)[/tex] using the above form formula.
[tex]P(D/A)=\frac{P(A\cap D)}{P(A)}[/tex] ;
[tex]P(A/D)=\frac{P(D\cap A)}{P(D)}[/tex]
From the given table, we have the values of P(A), P(D), [tex]P(D\cap A)[/tex] and [tex]P(A\cap D)[/tex].
Since, Probability=[tex]\frac{The number of wanted outcomes }{the number of possible outcomes}[/tex]
∴[tex]P(A)=\frac{8}{17}[/tex], [tex]P(D)=\frac{10}{17}[/tex], [tex]P(A\cap D)=\frac{2}{17}[/tex] and [tex]P(D\cap A)=\frac{2}{17}[/tex]
Now, putting these values in above formula we get,
[tex]P(D/A)=\frac{\frac{2}{17}} {\frac{8}{17}}[/tex]
[tex]P(D/A)=\frac{2} {8}=\frac{1}{4}[/tex]
[tex]P(D/A)= \frac{1}{4}[/tex].
[tex]P(A/D)=\frac{\frac{2}{17}} {\frac{10}{17}}=\frac{2}{10}[/tex]
[tex]P(A/D)=\frac{1}{5}[/tex]
As, you can see above that the values of P(A|D) and P(D|A) are not equal.
