I have two six-sided dice and I need to calculate the probability of achieving the following two scenarios.
- The sum of the dice is greater than or equal to 7 which is 7/12
- The second die is a 3 which is 1/6
The problem is calculating the intersection of 1 and 2. P(1 intersection 2), I know if you count the sample space, it's 1/12.
I've also checked that the 1 and 2 are not independent of each other - P(1) *P(2) != P(1intersection 2)
But how do I use the formula to calculate intersection without needing to count the sample space?
You have to count the sample space in some way or another. But there are ways so that you don't have to count the sample space of $36$, but rather a sample space of six two times.
The relevant formula is $$ P(A\cap B)=P(A)\cdot P(B\mid A) $$ Let $A$ be your 2. and $B$ be your 1, and you should be able to get the answer very quickly compared to how you did it, as each of the factors have a sample space of just $6$.