Two dice are rolled. A = First die is odd, B = sum of two dice is 9. Are A and B independent or disjoint? Find P(A and B), P(A or B)

Question

Two dice are rolled. $A$ is the event that the first die is odd and $B$ is the event that the sum of two dice is $9$. Are $A$ and $B$ independent or disjoint? Find $P(A \cap B), P(A \cup B)$.

I have $$P(A) = \frac{1}{2}, P(B) = \frac{4}{36}.$$ I calculated $P(A\cap B)$ as $P(A) \times P(B|A)$ to get $$P(A\cap B)=\frac{1}{2} *\frac{2}{36} = \frac{1}{36}.$$Then, I computed $P(A \cup B)$ as $P(A) + P(B) - P(A \cap B)$ and got $$P(A \cup B)=\frac{1}{2} + \frac{4}{36} - \frac{1}{36} = \frac{21}{36}.$$ I believe the events are dependent but the question seems to want independent or disjoint.

Answer 1

Your answers are incorrect!

1. There are two favourable events: $(3;6)$ and $(5;4)$

$$P(A\cap B)=\frac{2}{36}$$

2. As a direct consequence you have

$$P(A \cup B)= \frac{20}{36}$$

The two event are (correctly) NOT disjointed as $P(A \cap B)>0$

3. The two events are independent: in fact, by definition, two events $A,B$ are independent if and only if

$$P(A \cap B)=P(A)P(B)$$

or equivalently:

$P(B)=P(B|A)$

$P(A)=P(A|B)$

In your example, you have

$P(B)=\frac{4}{36}=\frac{1}{9}$

and $P(B|A)=\frac{P(A\cap B)}{P(A)}=\frac{1}{9}$