I am trying to learn conditional probability and cannot figure out what I'm doing wrong.
The problem:
We roll two dice. What is the conditional probability to have 1 on the first die under the condition that the sum of two numbers is 6?
My reasoning:
Total outcomes: $6^2$
The probability of the sum of two numbers is 6: we have 5 possibilities (1, 5), (2, 4), (3, 3), (4, 2), and (5, 1), which means $P[B] = \dfrac{5}{36}$.
Probability of the first die to have 1 is $P[A] = \dfrac{1}{6}$
Then, my reasoning is that $P[A \cap B] = \dfrac{5}{36} \dfrac{1}{6} = \dfrac{5}{216}$
Following: $P[A|B] = \dfrac{5}{216} / \dfrac{5}{36} = \dfrac{1}{36}$
Why is that wrong? What am I missing?
Here $P(A \text{ and } B) \not = P(A)P(B)$ as they are not independent events.
Instead say the only possibility for $A \text{ and } B$ is $(1,5)$, which has probability $\frac1{36}$, and then divide this by $\frac5{36}$ to give $P(A \mid B)= \frac15$.