Rolling a die - Probability of intersection of two events

Question

So I have a die, and two events.

$A = \{ \text{''Rolling an even number''} \}$

$B = \{ \text{ 4, 5, 6 } \}$

Then $\text{P}(A \cap B) = \text{P}(A|B)\cdot\text{P}(B) = \frac{2}{3} \cdot \frac{3}{6} = \frac{2}{6} = \frac{1}{3}$.

Or at least that is what I get? The lecturer says it is $\frac{2}{3}$ at the end but I do not see how. Can someone help?

Cheers

Answer 1

If the question asks

what is the probability of rolling an even number or rolling a number greater than 3?

you should use Inclusion-Exclusion Principle here. For 2 events $A$ and $B$, that is $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. So we have $$P(A \cup B) = \frac{1}{2} + \frac{1}{2} - \frac{1}{3} = \frac{2}{3}$$

Otherwise, if it asks

what is the probability of rolling an even number greater than 3?

then it is asking $P(A \cap B)$ and your solution is correct.