Dependent events
A and B are dependent events. This means that the outcome of B affects the outcome of A.
We have that "the probability that A occurs given that B occurs" is symbolized by
P(A|B)
This is what we want to find.
We have that:
P(A): probability that event A occurs.
P(A) = 1/4
P(B): probability that event B occurs.
P(B) = 8/9
P(A&B): probability that both events A and B occurs.
P(A&B) = 1/5
We have that:
[tex]\begin{gathered} P\mleft(A\&B\mright)=P\mleft(B\mright)\cdot P\mleft(A|B\mright) \\ \downarrow \\ \frac{P(A\&B)}{P(B)}=P(A|B) \end{gathered}[/tex]
Replacing in the equation:
[tex]\begin{gathered} \frac{P(A\&B)}{P(B)}=P(A|B) \\ \downarrow\text{ since }P\mleft(A\&B\mright)=\frac{1}{5}\text{ and }P\mleft(B\mright)=\frac{8}{9} \\ \frac{\frac{1}{5}}{\frac{8}{9}}=P(A|B) \end{gathered}[/tex]
Since,
[tex]\frac{\frac{1}{5}}{\frac{8}{9}}=\frac{1}{5}\cdot\frac{9}{8}=\frac{9}{40}[/tex]
Then
[tex]P(A|B)=\frac{9}{40}[/tex]
Answer: 9/40
