Answer:
[tex]P(A|B)=\frac{2}{3}[/tex]
[tex]P(A)*P(B)=\frac{1}{3}[/tex]
[tex]P(A) =\frac{2}{3}[/tex]
[tex]P(B) =\frac{1}{2}[/tex].
Step-by-step explanation:
We use the Venn diagram to calculate the desired probabilities.
Note that there are 6 possible results in the sample space
S = {1, 2, 3, 4, 5, 6}
Then note that in the region representing the intercept of A and B there are two possible values.
So
[tex]P (A\ and\ B) = \frac{2}{6} = \frac{1}{3}[/tex]
In the region that represents event A there are 4 possible outcomes {4, 5, 1, 2}
So
[tex]P(A) = \frac{4}{6} = \frac{2}{3}[/tex]
In the region that represents event B there are 3 possible outcomes {1, 2, 6}
So
[tex]P(B) = \frac{3}{6} = \frac{1}{2}[/tex].
Now
[tex]P(A | B)=\frac{P(A \ and\ B)}{P(B)}\\\\P(A | B)=\frac{\frac{1}{3}}{\frac{1}{2}}\\\\P(A|B)=\frac{2}{3}[/tex]
[tex]P(A)*P(B)=\frac{2}{3}*\frac{1}{2}=\frac{1}{3}[/tex]
