A die is rolled. What is the probability that the number is even and less than 4?
Event $A$: Numbers on a die that are even: 2, 4, 6
Event $B$: Numbers on a die that are less than 4: 1, 2, 3
There is only one number (2) that is in both events A and B.
Total outcomes $S$: Numbers on a die: 1, 2, 3, 4, 5, 6 (total = 6)
Ok, so obviously the one possible outcome in this example is $1/6$. But if I use the rule of multiplication which states that if the sample space is the same, which I think it is, then $P(a)P(b)$ should give me the right answer, which it doesn't: $1/4$.
(My logic here is that less than 4 is 50% chance and the even numbers are 50% chance).
So the big question is, why doesn't the multiplication rule work here?